hcju/isabelleps huggingface dataset:
dl1: fixes d :: "nat"   and g :: "'b"   and h :: "'b" shows "(if d = 1 then g else dichotomous_Lazard g h d) = dichotomous_Lazard g h d"
reflexive: fixes Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi" assumes "{(\<Psi>, P, P) |\<Psi> P. True} \<subseteq> Rel" shows "\<Psi> \<rhd> P \<leadsto>[Rel] P"
inducts: fixes p :: "'b \<times> 'c"   and x :: "'c"   and P :: "'c \<Rightarrow> bool" assumes "sndsp p x"   and "P (snd p)" shows "P x"
SeqAppendP_Mem_E: fixes H :: "fm set"   and f1 :: "tm"   and k1 :: "tm"   and f2 :: "tm"   and k2 :: "tm"   and g :: "tm"   and x :: "tm"   and y :: "tm"   and A :: "fm"   and u :: "name" assumes "H \<turnstile> SeqAppendP f1 k1 f2 k2 g"   and "insert (HPair x y IN f1) (insert (x IN k1) H) \<turnstile> A"   and "insert (HPair (Var u) y IN f2) (insert (HaddP k1 (Var u) x) (insert (Var u IN k2) H)) \<turnstile> A"   and "atom u \<sharp> (f1, k1, f2, k2, x, y, g, A)"   and "\<forall>C\<in>H. atom u \<sharp> C" shows "insert (HPair x y IN g) H \<turnstile> A"
tm_inc_err_range: fixes x :: "real"   and e :: "nat \<Rightarrow> real"   and t :: "taylor_model"   and i :: "float interval" assumes "x \<in>\<^sub>i range_tm e t + real_interval i" shows "x \<in>\<^sub>i range_tm e (tm_inc_err i t)"
B_authenticates_A: fixes K :: "nat"   and A :: "agent"   and Ta :: "nat"   and evs :: "event list"   and B :: "agent"   and Tk :: "nat" assumes "Crypt K \<lbrace>Agent A, Number Ta\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt (shrK B) \<lbrace>Number Tk, Agent A, Key K\<rbrace> \<in> parts (knows Spy evs)"   and "Key K \<notin> analz (knows Spy evs)"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> bankerberos" shows "Says A B \<lbrace>Crypt (shrK B) \<lbrace>Number Tk, Agent A, Key K\<rbrace>, Crypt K \<lbrace>Agent A, Number Ta\<rbrace>\<rbrace> \<in> set evs"
T2: shows "\<lfloor>\<P> (\<lambda>x w. \<forall>xa. (\<P> xa \<^bold>\<rightarrow> xa x) w)\<rfloor>"
nlta__lea: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "\<not> A B C LtA D E F"   and "A \<noteq> B"   and "B \<noteq> C"   and "D \<noteq> E"   and "E \<noteq> F" shows "D E F LeA A B C"
oexp_def: shows "oexp = {(f, g). f \<in> FINFUNC \<and> g \<in> FINFUNC \<and> ((let m = s.max_fun_diff f g in (f m, g m) \<in> r) \<or> f = g)}"
solution_solves_ode: shows "(solution solves_ode f) (cball t et) (cball (flow t0 x0 t) ex)"
progress: fixes \<Phi> :: "String.literal \<Rightarrow> String.literal \<Rightarrow> (ty list \<times> ty err list) option list"   and t :: "'thread_id"   and xcp :: "'addr option"   and h :: "'heap"   and f :: "'addr frame"   and frs :: "'addr frame list" assumes "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"   and "\<Phi> \<turnstile> t:(xcp, h, f # frs) \<surd>" shows "\<exists>ta \<sigma>'. P,t \<turnstile> (xcp, h, f # frs) -ta-jvm\<rightarrow> \<sigma>'"
assms_2: shows "\<forall>rel\<in>R. finite (Atoms rel)"
Joints2_def: fixes XS :: "('b \<times> 'b pdevs) list"   and Y :: "'c \<times> 'c pdevs" shows "Joints2 XS Y = valuate (\<lambda>e. (map (aform_val e) XS, aform_val e Y))"
Ctxt_lowerm: fixes n :: "nat"   and G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and m :: "nat" assumes " \<Turnstile>\<^sub>n G  "   and "m < n" shows " \<Turnstile>\<^sub>m G  "
not_less_empty: fixes M :: "'k multiset"   and r :: "('k \<times> 'k) set" shows "(M, {#}) \<notin> mult1 r"
for_insert_elt_correct: fixes l :: "nat"   and h :: "heap"   and a :: "'b array" assumes "l \<le> Array.length h a"   and "1 \<le> l" shows "\<exists>h'. effect (for [1..<l] (insert_elt a)) h h' () \<and> Array.length h a = Array.length h' a \<and> sorted (take l (Array.get h' a)) \<and> mset (Array.get h a) = mset (Array.get h' a)"
mantissa_nonneg_iff: fixes x :: "float" shows "(0 \<le> mantissa x) = (0 \<le> x)"
vertexI1: fixes \<Gamma> :: "('k, 'l) graph_scheme"   and x :: "'k"   and y :: "'k" assumes "edge \<Gamma> x y" shows "vertex \<Gamma> x"
assms: fixes x :: "'b"   and y :: "'b" assumes "x \<in> A"   and "y \<in> A"   and "x \<noteq> y" shows "coprime (g x) (g y)"
dvd_mod_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "'b"   and b :: "'b"   and a :: "'b" assumes "class.semidom_modulo divide plus minus zero times one modulo"   and "dvd.dvd times c b" shows "dvd.dvd times c (modulo a b) = dvd.dvd times c a"
distinct_Ex1: fixes xs :: "'b list"   and x :: "'b" assumes "cycle xs"   and "x \<in> set xs" shows "\<exists>!i. i < length xs \<and> xs ! i = x"
ring_distribs_3: fixes a :: "'d"   and b :: "'d"   and c :: "'d" shows "(a - b) * c = a * c - b * c"
IH_1: shows "\<forall>t1 t2. t1 \<cdot> t2 \<prec> s1 \<longrightarrow> t1 \<prec> s1 \<and> t2 \<prec> s1"
dsjCont: fixes A :: "fmla" shows "boolean_axiomsp (A OR A IMP A)"
vbrelationI: fixes r :: "V" assumes "vpairs r = r" shows "vbrelation r"
update_def: fixes r :: "'b ref"   and v :: "'b" shows "r := v = Heap_Monad.heap (\<lambda>h. ((), Ref.set r v h))"
infdist_le_hausdorff_distance: fixes x :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "x \<in> A"   and "bounded A"   and "bounded B" shows "infdist x B \<le> hausdorff_distance A B"
case_distrib: fixes h :: "'h \<Rightarrow> 'g"   and f :: "'i \<Rightarrow> 'j \<Rightarrow> 'h"   and prod :: "'i \<times> 'j" shows "h (case prod of (x, xa) \<Rightarrow> f x xa) = (case prod of (x1, x2) \<Rightarrow> h (f x1 x2))"
lipschitz: fixes t :: "real" assumes "t \<in> T" shows "B'-lipschitz_on X (f t)"
elims_2: fixes x :: "'a extended"   and xa :: "'a extended"   and xb :: "('a, 'b option) insert_message" assumes "is_concurrent x xa xb" obtains pm :: "'a extended" and sm :: "'a extended" and x :: "('a, 'b option) insert_message" where "xa__ = pm" and "xa = sm" and "xb = x" and "x \<in> set s \<and> Psi.subset (a pm, a sm) (a (P x), a (S x)) \<and> elem (a \<lbrakk>I x\<rbrakk>) (a pm, a sm)"
has_disc_negated_alist_and': fixes disc :: "'b \<Rightarrow> bool"   and neg :: "bool"   and as :: "'b negation_type list" shows "has_disc_negated disc neg (alist_and' as) = (\<exists>a\<in>set as. has_disc_negated disc neg (negation_type_to_match_expr a))"
helper: fixes A :: "int"   and B :: "int" shows "A * B + B = (A + 1) * B"
evaln_SkipD: fixes G :: "prog"   and s :: "state"   and n :: "nat"   and s' :: "state" assumes "G\<turnstile>s \<midarrow>Skip\<midarrow>n\<rightarrow> s'" shows "s' = s"
complex_inner_1_right: fixes x :: "complex" shows "x \<bullet> 1 = Re x"
simps_1: fixes edge_rev :: "'b \<Rightarrow> 'b"   and edge_succ :: "'b \<Rightarrow> 'b"   and more :: "'c" shows "edge_rev \<lparr>edge_rev = edge_rev, edge_succ = edge_succ, \<dots> = more\<rparr> = edge_rev"
diff_pCons: fixes a :: "'a"   and p :: "'a poly"   and b :: "'a"   and q :: "'a poly" shows "pCons a p - pCons b q = pCons (a - b) (p - q)"
cobounded2: fixes a :: "'a::semilattice_inf"   and b :: "'a::semilattice_inf" shows "(a::'a::semilattice_inf) \<sqinter> (b::'a::semilattice_inf) \<le> b"
Inf_lower2: fixes u :: "'a"   and A :: "'a set"   and v :: "'a" assumes "u \<in> A"   and "u \<le> v" shows "Inf A \<le> v"
exec_n_step: fixes n :: "int"   and n' :: "int"   and P :: "instr list"   and stk :: "char list \<Rightarrow> int"   and s :: "int list"   and k :: "nat"   and stk' :: "char list \<Rightarrow> int"   and s' :: "int list" assumes "n \<noteq> n'" shows "P \<turnstile> (n, stk, s) \<rightarrow>^k (n', stk', s') = (\<exists>c. P \<turnstile> (n, stk, s) \<rightarrow> c \<and> P \<turnstile> c \<rightarrow>^(k - 1) (n', stk', s') \<and> 0 < k)"
simps_1: fixes \<psi> :: "'b ltln"   and X :: "'b ltln set" shows "(X\<^sub>n \<psi>)[X]\<^sub>\<nu> = X\<^sub>n \<psi>[X]\<^sub>\<nu>"
wf_all: fixes \<Gamma> :: "binding list"   and T :: "POPLmark.type"   and U :: "POPLmark.type" assumes "\<Gamma> \<turnstile>\<^sub>w\<^sub>f T"   and "TVarB T \<Colon> \<Gamma> \<turnstile>\<^sub>w\<^sub>f U" shows "\<Gamma> \<turnstile>\<^sub>w\<^sub>f (\<forall><:T. U)"
lm111: fixes Z :: "'b set"   and X :: "'b set" assumes "finite Z" shows "sum (chi X) Z = card (X \<inter> Z)"
induct: fixes x :: "'c set"   and P :: "'c set \<Rightarrow> bool" assumes "finite x"   and "P {}"   and "\<And>A a. \<lbrakk>finite A; P A\<rbrakk> \<Longrightarrow> P (insert a A)" shows "P x"
one_dim_norm: fixes x :: "'c" shows "norm x = cmod (one_dim_iso x)"
inv_rule: fixes T :: "('e, 'f) TS_scheme"   and I :: "'e set"   and P :: "'e set" assumes "init T \<subseteq> I"   and "I \<subseteq> P"   and "{I} TS.trans T {> I}" shows "reach T \<subseteq> P"
boundedE: fixes b :: "'h"   and c :: "'h"   and a :: "'h" assumes "sup b c \<le> a" obtains "b \<le> a" and "c \<le> a"
cases: fixes x :: "'c option \<times> 'd option" obtains xa :: "'c" and y :: "'d" where "x = (Some xa, Some y)"   | "x = (None, None)"   | v :: "'d" where "x = (None, Some v)"   | va :: "'c" where "x = (Some va, None)"
distinct_enqueue_new_threads: fixes ts :: "'a \<Rightarrow> ('b \<times> 'c \<Rightarrow>f nat) option"   and ntas :: "('a, 'b, 'd) new_thread_action list"   and queue :: "'a list" assumes "FWThread.thread_oks ts ntas"   and "set queue = dom ts"   and "distinct queue" shows "distinct (Round_Robin.enqueue_new_threads queue ntas)"
cases: fixes a1 :: "'c"   and a2 :: "'d list"   and a3 :: "'c"   and t :: "('c \<times> 'd \<times> 'c) set" assumes "(a1, a2, a3) \<in> trcl t" obtains c :: "'c" where "a1 = ca__" and "a2 = []" and "a3 = ca__"   | c :: "'c" and a :: "'d" and c' :: "'c" and w :: "'d list" and c'' :: "'c" where "a1 = c" and "a2 = a # w" and "a3 = c''" and "(c, a, c') \<in> t" and "(c', w, c'') \<in> trcl t"
coprime_1_left: fixes a :: "'a" shows "coprime (1::'a) a"
is_abelian_subgroup: shows "abelian_subgroup (a_kernel R S h) R"
glbound_in_ofD_in: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a set"   and x :: "'a"   and y :: "'a"   and b :: "'a" assumes "ordering less_eq less"   and "ordering.glbound_in_of less_eq P x y b" shows "b \<in> P"
reduction_ample: fixes q :: "'state" assumes "q \<in> nodes" shows "ample_set q {a. ren a q}"
BadRead_def: fixes mm :: "state \<Rightarrow> Locs \<Rightarrow> Vals"   and rs :: "state \<Rightarrow> PrIds \<Rightarrow> Vals"   and p :: "PrIds"   and l :: "Locs" shows "BadRead mm rs p l \<equiv> #l \<notin> #MemLoc \<and> (rs!p)$ = #BadArg"
COCOUNT_def: shows "COCOUNT = sigma UNIV {{x} |x. True}"
map_le_def: fixes m\<^sub>1 :: "'a \<Rightarrow> 'b option"   and m\<^sub>2 :: "'a \<Rightarrow> 'b option" shows "(m\<^sub>1 \<subseteq>\<^sub>m m\<^sub>2) = (\<forall>a\<in>dom m\<^sub>1. m\<^sub>1 a = m\<^sub>2 a)"
assms: shows "WT_rest \<I>_adv \<I>_usr I rest"
esqrt_of_neg: fixes x :: "ereal" assumes "x \<le> 0" shows "esqrt x = 0"
IH: fixes ys :: "'a list" assumes "concat xss = ys @ zs"   and "xss \<noteq> []" shows "\<exists>xss1 xs xs' xss2. xss = xss1 @ (xs @ xs') # xss2 \<and> ys = concat xss1 @ xs \<and> zs = xs' @ concat xss2"
relcompI: fixes a :: "'c"   and b :: "'d"   and r :: "('c \<times> 'd) set"   and c :: "'e"   and s :: "('d \<times> 'e) set" assumes "(a, b) \<in> r"   and "(b, c) \<in> s" shows "(a, c) \<in> r O s"
coeff_in_carrier: fixes p :: "'a list"   and i :: "nat" assumes "set p \<subseteq> carrier R" shows "R.coeff p i \<in> carrier R"
MClkFwd_ch_enabled: fixes send :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and cst :: "state \<Rightarrow> PrIds \<Rightarrow> mClkState"   and p :: "PrIds" shows "\<turnstile> Enabled MClkFwd send rcv cst p \<longrightarrow> Enabled (<MClkFwd send rcv cst p>_(cst!p, rtrner send!p, caller rcv!p))"
P1_real_ideal_DDH_advantage_false_bound: fixes \<sigma> :: "bool"   and n :: "nat"   and M :: "'grp \<times> 'grp"   and z :: "'d"   and \<A> :: "('grp \<times> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'd \<Rightarrow> (('grp \<times> 'grp \<times> 'grp) \<times> 'e) spmf) \<times> ('grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<times> 'grp \<Rightarrow> 'e \<Rightarrow> ((('grp \<times> 'grp) \<times> 'grp \<times> 'grp) \<times> 'e) spmf) \<times> ('e \<Rightarrow> 'f spmf)"   and D :: "'f \<times> 'grp \<Rightarrow> bool spmf" assumes "\<sigma> = False" shows "mal_def.adv_P1 n M \<sigma> z (P1_S1 n, P1_S2) \<A> D \<le> local.ddh.advantage n (P1_DDH_mal_adv_\<sigma>_false n M z \<A> D) + local.ddh.advantage n (local.ddh.DDH_\<A>' n (P1_DDH_mal_adv_\<sigma>_false n M z \<A> D))"
assms: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "f (f x y) z = f x (f y z)"
minimum_spanning_tree_def: fixes F :: "('c, 'd) graph"   and G :: "('c, 'd) graph" shows "minimum_spanning_tree F G \<equiv> spanning_tree F G \<and> optimal_tree F G"
inj1': shows "inj_on (\<lambda>xs. {xs} // {\<cong>}) (map \<phi> ` set Fs\<^sub>1)"
a_assoc: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a"   and z :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "y \<in> carrier P"   and "z \<in> carrier P" shows "x \<oplus>\<^bsub>P\<^esub> y \<oplus>\<^bsub>P\<^esub> z = x \<oplus>\<^bsub>P\<^esub> (y \<oplus>\<^bsub>P\<^esub> z)"
measurable_funpow_T': fixes n :: "nat" shows "T ^^ n \<in> tail_algebra n \<rightarrow>\<^sub>M P"
assms_13: shows "InvariantClCurrentLevel (getCl state) (getM state)"
insert_Times_insert: fixes a :: "'a"   and A :: "'a set"   and b :: "'b"   and B :: "'b set" shows "insert a A \<times> insert b B = insert (a, b) (A \<times> insert b B \<union> insert a A \<times> B)"
gcd_add_dvd_right1: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a dvd b" shows "gcd a (b + c) = gcd a c"
prg_cert_def: fixes P :: "jvm_prog"   and phi :: "char list \<Rightarrow> char list \<Rightarrow> (ty list \<times> ty err list) option list"   and C :: "char list"   and mn :: "char list" shows "prg_cert P phi C mn \<equiv> let (C, b, Ts, T\<^sub>r, mxs, mxl\<^sub>0, ins, et) = method P C mn in mk_cert P mxs T\<^sub>r et ins (phi C mn)"
lrep: shows "lb \<le> r \<and> (fst ([\<theta>]<>\<nu>) \<equiv>\<^sub>E lb)"
IsoInvId2: fixes C :: "('a, 'b, 'c) Category_scheme"   and f :: "'b" assumes "Category C"   and "ciso\<^bsub>C\<^esub> f" shows "f ;;\<^bsub>C\<^esub> Cinv\<^bsub>C\<^esub> f = id\<^bsub>C\<^esub> (dom\<^bsub>C\<^esub> f)"
Fc: shows "Ftor NTCod (YFtorNT' C f) : Op C \<longrightarrow> SET"
greaterThanAtMost_subseteq_atLeastLessThan_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.dense_linorder less_eq less" shows "(ord.greaterThanAtMost less_eq less a b \<subseteq> ord.atLeastLessThan less_eq less c d) = (less a b \<longrightarrow> less_eq c a \<and> less b d)"
SOME_Basis: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and Basis :: "'b set" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "(SOME i. i \<in> Basis) \<in> Basis"
update_S_def: fixes s_val :: "word1"   and psr_val :: "word32" shows "update_S s_val psr_val \<equiv> let tmp0 = psr_val AND 4294967167 in tmp0 OR (ucast s_val << 7)"
pn_map_def: fixes nds :: "('a, 'b) node_scheme set" shows "pn_map nds \<equiv> FOREACH nds (\<lambda>nd m. pn_props (old nd) \<bind> (\<lambda>PN. RETURN (m(name nd \<mapsto> PN)))) Map.empty"
j0: shows "\<forall>j\<in>{1..<n}. H $$ (0, j) = (0::'a)"
assms: shows "R[x\<turnstile>n>y] = AndL1 v.N u"
A_0_imp_Gauss_Jordan_0: fixes A :: "(('b, 'c) vec, 'd) vec" assumes "A = 0" shows "Gauss_Jordan A = 0"
leq_mix_imp_weak_inferior: fixes \<alpha> :: "real"   and \<beta> :: "real" assumes "local.some_best \<succeq>[\<R>] local.some_worst \<and> (local.some_worst, local.some_best) \<notin> \<R>"   and "\<alpha> \<in> {0..1}"   and "\<beta> \<in> {0..1}"   and "mix_pmf \<beta> local.some_best local.some_worst \<succeq>[\<R>] mix_pmf \<alpha> local.some_best local.some_worst" shows "\<alpha> \<le> \<beta>"
lfp_def: fixes Inf :: "'g set \<Rightarrow> 'g"   and Sup :: "'g set \<Rightarrow> 'g"   and inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and bot :: "'g"   and top :: "'g"   and f :: "'g \<Rightarrow> 'g" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "complete_lattice.lfp Inf less_eq f = Inf {u. less_eq (f u) u}"
nth_drop: fixes n :: "nat"   and xs :: "'b list"   and i :: "nat" assumes "n \<le> length xs" shows "drop n xs ! i = xs ! (n + i)"
axioms_3: fixes X :: "'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 \<FF>' :: "'d set \<Rightarrow> 'f set"   and \<rho>' :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'f \<Rightarrow> 'f"   and b' :: "'f"   and add_str' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and mult_str' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero_str' :: "'d set \<Rightarrow> 'f"   and one_str' :: "'d set \<Rightarrow> 'f"   and fam_morphisms :: "'d set \<Rightarrow> 'e \<Rightarrow> 'f" assumes "morphism_presheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' fam_morphisms" shows "morphism_presheaves_of_rings_axioms is_open \<FF> \<rho> add_str mult_str zero_str one_str \<FF>' \<rho>' add_str' mult_str' zero_str' one_str' fam_morphisms"
is_bezout_ext_def: fixes 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 bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" assumes "class.bezout_ring times one plus zero minus uminus" shows "bezout_ring.is_bezout_ext times plus uminus bezout = (\<forall>a b. let (p, q, u, v, gcd_a_b) = bezout a b in plus (times p a) (times q b) = gcd_a_b \<and> dvd.dvd times gcd_a_b a \<and> dvd.dvd times gcd_a_b b \<and> (\<forall>d'. dvd.dvd times d' a \<and> dvd.dvd times d' b \<longrightarrow> dvd.dvd times d' gcd_a_b) \<and> times gcd_a_b u = uminus b \<and> times gcd_a_b v = a)"
simps_1: fixes d :: "'dyn" shows "box_operand (OpDyn d) = OpDyn d"
exp_times_self_mono: fixes x :: "real"   and y :: "real" assumes "- 1 \<le> x"   and "x \<le> y" shows "x * exp x \<le> y * exp y"
more\<^sub>L_def: shows "mylens.more\<^sub>L = \<lparr>lens_get = mylens.more, lens_put = \<lambda>\<sigma> u. \<sigma>\<lparr>mylens.more := u\<rparr>\<rparr>"
DSTEP_star_comp1: fixes l :: "('c + 'd) list"   and r :: "('c + 'd) list"   and t :: "('c + 'd) list"   and G :: "('c \<times> ('c, 'd) RHS) list" assumes "l \<cdot> r -G\<rightarrow>\<^sup>* t" shows "\<exists>l' r'. l -G\<rightarrow>\<^sup>* l' \<and> r -G\<rightarrow>\<^sup>* r' \<and> t = l' \<cdot> r'"
dbm_entry_val_add_2: fixes u :: "'c \<Rightarrow> 'd"   and c :: "'c"   and a :: "'d DBMEntry"   and d :: "'c"   and b :: "'d DBMEntry" assumes "dbm_entry_val u None (Some c) a"   and "dbm_entry_val u (Some c) (Some d) b" shows "dbm_entry_val u None (Some d) (dbm_add a b)"
Nodes_in_pret: fixes t :: "dag"   and levellista :: "ref list list"   and var :: "ref \<Rightarrow> nat"   and nb :: "nat" assumes "wf_ll t levellista var"   and "nb \<le> length levellista" shows "Nodes nb levellista \<subseteq> set_of t"
R'_respect: shows "(\<lambda>xa. {per_union R x y `` {xa}}) respects R"
phull_closed_mult_scalar: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and p :: "'a \<Rightarrow>\<^sub>0 'd"   and B :: "('a \<Rightarrow>\<^sub>0 'd) set"   and c :: "'d" assumes "term_powerprod pair_of_term term_of_pair"   and "p \<in> phull B" shows "term_powerprod.mult_scalar pair_of_term term_of_pair (monomial c (0::'b)) p \<in> phull B"
monofun_cfun_fun: fixes f :: "'a \<rightarrow> 'b"   and g :: "'a \<rightarrow> 'b"   and x :: "'a" assumes "f \<sqsubseteq> g" shows "f\<cdot>x \<sqsubseteq> g\<cdot>x"
iterates_above_triangle: fixes x :: "'a"   and a :: "'a"   and y :: "'a" assumes "x \<in> iterates_above a"   and "y \<in> iterates_above a"   and "a \<in> Field leq" shows "x \<in> iterates_above y \<or> y \<in> iterates_above x"
disc_2: fixes sys :: "'a \<Rightarrow> ('b, 'a) process_term"   and T :: "('b, 'a) process_term" assumes "\<not> isACT sys T" shows "isChoice (solution sys T)"
Fun_subterm_inside_params: fixes t :: "('a, 'b) Term.term"   and f :: "'a"   and X :: "('a, 'b) Term.term list" shows "(t \<sqsubset> Fun f X) = (t \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t set X)"
u_2: shows "u \<longlonglongrightarrow> 0"
eventually_subst': fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<forall>\<^sub>F x in F. f x = g x" shows "(\<forall>\<^sub>F x in F. P x (f x)) = (\<forall>\<^sub>F x in F. P x (g x))"
iterations_spec_def: shows "iterations_spec = SPEC (\<lambda>n. True)"
2: shows "is_left_adjoint (tab\<^sub>1 r \<star> (tab\<^sub>0 r)\<^sup>*)"
least_cong: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and x' :: "'c"   and A :: "'c set" assumes "weak_partial_order L"   and "x .=\<^bsub>L\<^esub> x'"   and "x \<in> carrier L"   and "x' \<in> carrier L"   and "is_closed\<^bsub>L\<^esub> A" shows "least L x A = least L x' A"
r_consist_upto_1: fixes r_psi :: "recf"   and j :: "nat"   and i :: "nat"   and e :: "nat" assumes "recfn 2 r_psi"   and "\<forall>k<j. eval r_psi [i, k] \<down>" shows "eval (r_consist_upto r_psi) [j, i, e] = (if \<forall>k<j. eval r_psi [i, k] \<down>= e_nth e k then Some 0 else Some 1)"
SeqQuote_def: fixes x :: "hf"   and x' :: "hf"   and s :: "hf"   and k :: "hf" shows "SeqQuote x x' s k \<equiv> BuildSeq2 (\<lambda>y y'. y = 0 \<and> y' = 0) (\<lambda>u u' v v' w w'. u = v \<triangleleft> w \<and> u' = q_Eats v' w') s k x x'"
assms_2: shows "t \<in> set_pmf (random_bst A)"
eq_fold: fixes M :: "'a multiset" shows "\<Prod>\<^sub># M = fold_mset (*) (1::'a) M"
sets_eq: fixes x :: "'a" assumes "x \<in> space M" shows "sets (f x \<Otimes>\<^sub>M g x) = sets (N \<Otimes>\<^sub>M L)"
dims_tensor_from_lookup: fixes ds :: "nat list"   and e :: "nat list \<Rightarrow> 'm" shows "dims (tensor_from_lookup ds e) = ds"
int_zero_eq: shows "\<zero>\<^bsub>\<Z>\<^esub> = 0"
cod_src_eq_dom_src: shows "\<nu>.cod.src = \<nu>.dsrc"
conF: fixes n :: "nat" shows "continuous_on UNIV (F n)"
exact_seq_length2: fixes t :: "'b monoid list \<times> ('b \<Rightarrow> 'b) list" assumes "exact_seq t" shows "Suc 0 \<le> length (snd t)"
psubstEnv_idEnv_id: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option" assumes "goodEnv rho" shows "rho &[idEnv] = rho"
rep_nat_pp_pp_eq_iff: fixes x :: "('a, 'b) pp"   and y :: "('a, 'b) pp" shows "(rep_nat_pp x = rep_nat_pp y) = (x = y)"
return_transfer: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "state_dp_consistency.rel_fun_lifted R R Wrap State_Monad.return"