hcju/isabelleps huggingface dataset:
collision_react_def: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and s\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and s\<^sub>o :: "real"   and \<delta> :: "real"   and time_set :: "real set" assumes "safe_distance_normal a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta>" shows "safe_distance_normal.collision_react a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta> time_set \<equiv> \<exists>t\<in>time_set. safe_distance_normal.u a\<^sub>e v\<^sub>e s\<^sub>e \<delta> t = movement.s a\<^sub>o v\<^sub>o s\<^sub>o t"
Gamma_series'_minus_of_nat: fixes k :: "nat" shows "\<forall>\<^sub>F n in sequentially. Gamma_series' (- of_nat k) n = (0::'a)"
sig_red_zero_is_red: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and r :: "'t \<Rightarrow>\<^sub>0 'b" assumes "sig_red_zero sing_reg F r"   and "rep_list r \<noteq> 0" shows "is_sig_red sing_reg (\<preceq>) F r"
border_short_dec: fixes x :: "'b list"   and w :: "'b list" assumes "x \<le>b w"   and "\<^bold>|x\<^bold>| + \<^bold>|x\<^bold>| \<le> \<^bold>|w\<^bold>|" shows "x \<cdot> x\<inverse>\<^sup>>w\<^sup><\<inverse>x  \<cdot> x = w"
instance_of\<^sub>l\<^sub>s_def: fixes C\<^sub>1 :: "fterm literal set"   and C\<^sub>2 :: "fterm literal set" shows "instance_of\<^sub>l\<^sub>s C\<^sub>1 C\<^sub>2 = (\<exists>\<sigma>. C\<^sub>1 = C\<^sub>2 \<cdot>\<^sub>l\<^sub>s \<sigma>)"
pointwiseI: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'b \<Rightarrow> 'c"   and m' :: "'b \<Rightarrow> 'd" assumes "\<And>x. P (m x) (m' x)" shows "pointwise P m m'"
line_integral_dist: shows "one_chain_line_integral F {i, j} \<gamma> = one_chain_line_integral F {i} \<gamma> + one_chain_line_integral F {j} \<gamma>"
is_gcd_some_gcdI: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "\<exists>x. dvd.is_gcd times x a b" shows "dvd.is_gcd times (dvd.some_gcd times a b) a b"
lhd_llist_inf: fixes xs :: "'b llist"   and ys :: "'b llist" assumes "xs \<noteq> LNil"   and "ys \<noteq> LNil"   and "lhd xs = lhd ys" shows "lhd (xs \<sqinter> ys) = lhd ys"
nn_integral_subalgebra: fixes f :: "'b \<Rightarrow> ennreal"   and N :: "'b measure"   and M :: "'b measure" assumes "f \<in> borel_measurable N"   and "sets N \<subseteq> sets M"   and "space N = space M"   and "\<And>A. A \<in> sets N \<Longrightarrow> emeasure N A = emeasure M A" shows "integral\<^sup>N N f = integral\<^sup>N M f"
lookup_HSem_other: fixes y :: "'b"   and h :: "('b \<times> 'a) list"   and \<rho> :: "'b \<Rightarrow> 'c" assumes "y \<notin> domA h" shows "(\<lbrace> h \<rbrace>\<rho>) y = \<rho> y"
well_base_filter: fixes M :: "'a multiset"   and p :: "'a \<Rightarrow> bool" assumes "local.well_base M" shows "local.well_base (filter_mset p M)"
tm_wf_tcopy_2: shows "tm_wf0 tcopy_loop"
indep_sets_pullback_algebra: shows "local.P.indep_sets (\<lambda>i. sets.pullback_algebra M (\<lambda>f. f i) (space P)) UNIV"
assms_2: shows "f (0::'b) = (0::'a)"
section_retraction_compose: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and e :: "'b"   and m :: "'b"   and e' :: "'b"   and m' :: "'b" assumes "category C"   and "partial_magma.ide C (C e m)"   and "partial_magma.ide C (C e' m')"   and "partial_magma.arr C (C m' m)" shows "partial_magma.ide C (C (C e e') (C m' m))"
simps_27: fixes f1 :: "b_e \<Rightarrow> 'a"   and f2 :: "'a"   and f3 :: "cl \<Rightarrow> 'a"   and f4 :: "nat \<Rightarrow> e list \<Rightarrow> e list \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> nat \<Rightarrow> v list \<Rightarrow> e list \<Rightarrow> 'a"   and x3 :: "cl" shows "(case Callcl x3 of $x \<Rightarrow> f1 x | Trap \<Rightarrow> f2 | Callcl x \<Rightarrow> f3 x | Label x xa xb \<Rightarrow> f4 x xa xb | Local x xa xb xc \<Rightarrow> f5 x xa xb xc) = f3 x3"
reach_is_node: fixes A :: "nat bdd list \<times> bool list"   and n :: "nat"   and p :: "nat"   and w :: "bool list list"   and q :: "nat" assumes "aut_dfa A n"   and "dfa_reach A p w q"   and "dfa_is_node A p"   and "list_all (is_alph n) w" shows "dfa_is_node A q"
assms_4: shows "\<not> is_zero_row_upt_k (i + (1::'c)) k A"
prefix_singleton_TT: fixes x :: "'r"   and ys :: "[:'r:]" shows "(prefix\<cdot>[:x:]\<cdot>ys = TT) = (x \<noteq> \<bottom> \<and> (\<exists>zs. zs \<noteq> \<bottom> \<and> ys = x :# zs))"
exe_consistent_length_tcsigs_def: fixes arss :: "('a \<times> ('b \<times> 'c list) list) list" shows "exe_consistent_length_tcsigs arss \<equiv> \<forall>ars\<in>snd ` set arss. \<forall>ss\<^sub>1\<in>snd ` set ars. \<forall>ss\<^sub>2\<in>snd ` set ars. length ss\<^sub>1 = length ss\<^sub>2"
octo_of_real_times_commute: fixes r :: "real"   and q :: "octo" shows "octo_of_real r * q = q * octo_of_real r"
permute_lift_def: fixes p :: "perm"   and x :: "'a lift" shows "p \<bullet> x = (case x of \<bottom> \<Rightarrow> \<bottom> | Def x \<Rightarrow> Def (p \<bullet> x))"
diff_add: fixes a :: "'a multiset"   and b :: "'a multiset" assumes "a \<subseteq># b" shows "b - a + a = b"
Adm_to_En: fixes \<beta> :: "'e list"   and \<rho> :: "'e V_rec \<Rightarrow> 'e set"   and e :: "'e" assumes "\<beta> \<in> Tr\<^bsub>(induceES SES)\<^esub>"   and "Adm \<V> \<rho> Tr\<^bsub>(induceES SES)\<^esub> \<beta> e" shows "\<exists>s\<in>S\<^bsub>SES\<^esub>. s0\<^bsub>SES\<^esub> \<beta>\<Longrightarrow>\<^bsub>SES\<^esub> s \<and> En \<rho> s e"
assms_2: fixes i :: "'a" assumes "i \<in> I" shows "f i = g i"
w_simps_4: shows "trg \<mu>.chine = src (tab\<^sub>0 (local.dom \<tau>))"
order_trans: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "class.preorder less_eq less"   and "less_eq x y"   and "less_eq y z" shows "less_eq x z"
negatively_invariant_iff: fixes M :: "'a set" shows "negatively_invariant M = ((\<Union>x\<in>M. flow0 x ` (existence_ivl0 x \<inter> {..0})) \<subseteq> M)"
nonneg_Reals_cmod_eq_Re: fixes z :: "complex" assumes "z \<in> \<real>\<^sub>\<ge>\<^sub>0" shows "cmod z = Re z"
skel_asTerm_qSkel: fixes qX :: "('a, 'b, 'varSort, 'var, 'c) qTerm" assumes "qGood qX" shows "skel (asTerm qX) = qSkel qX"
assms: shows "low_equal s1 s2 \<and> t1 = snd (fst (set_annul True (snd (fst (write_cpu (cpu_reg_val nPC s2 + 4) nPC (snd (fst (write_cpu (cpu_reg_val nPC s2) PC s1)))))))) \<and> t2 = snd (fst (set_annul True (snd (fst (write_cpu (cpu_reg_val nPC s2 + 4) nPC (snd (fst (write_cpu (cpu_reg_val nPC s2) PC s2))))))))"
T0: shows "\<forall>x\<in>carrier. \<forall>y\<in>carrier. x \<noteq> y \<longrightarrow> (\<exists>u\<in>nhds x. y \<notin> u) \<or> (\<exists>v\<in>nhds y. x \<notin> v)"
c_graph_lm_3: fixes x :: "nat"   and y :: "nat"   and f :: "nat \<Rightarrow> nat" shows "(c_pair x y \<in> c_graph f) = (y = f x)"
exhaust: fixes y :: "'b llist" obtains "y = LNil"   | x21 :: "'b" and x22 :: "'b llist" where "y = x21 ## x22"
map_simp_ide: fixes a :: "'a \<times> 'a" assumes "CC.ide a" shows "\<sigma>.map a = \
[fst a, snd a]"
one_Gp_one: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set" assumes "Group G"   and "G \<guillemotright> H " shows "\<one>\<^bsub>\<natural>\<^bsub>G\<^esub>H\<^esub> = \<one>\<^bsub>G\<^esub>"
fine_single: fixes a :: "real"   and b :: "real"   and x :: "real"   and \<delta> :: "real \<Rightarrow> real" assumes "a < b"   and "a \<le> x"   and "x \<le> b"   and "b - a < \<delta> x" shows "fine \<delta> (a, b) [(a, x, b)]"
FUadmitFO_def: fixes \<sigma> :: "'b \<Rightarrow> ('c, 'd) trm"   and \<theta> :: "('c + 'b, 'e, 'd) formula"   and U :: "('d + 'd) set" shows "FUadmitFO \<sigma> \<theta> U = ((\<Union>i\<in>{i. Inl (Inr i) \<in> SIGF \<theta>}. FVT (\<sigma> i)) \<inter> U = {})"
CFG_wf_axioms: shows "CFG_wf sourcenode targetnode kind valid_edge (_Entry_) get_proc get_return_edges procs Main Def Use ParamDefs ParamUses"
sscanl_ww_fusion: fixes wrap :: "'s \<rightarrow> 'r"   and unwrap :: "'r \<rightarrow> 's"   and f :: "'r \<rightarrow> 't \<rightarrow> 'r"   and f' :: "'s \<rightarrow> 't \<rightarrow> 's"   and z :: "'r" assumes "wrap oo unwrap = ID"   and "\<And>z x. x \<noteq> \<bottom> \<Longrightarrow> unwrap\<cdot>(f\<cdot>(wrap\<cdot>z)\<cdot>x) = f'\<cdot>(unwrap\<cdot>(wrap\<cdot>z))\<cdot>x" shows "sscanl\<cdot>f\<cdot>z = smap\<cdot>wrap oo sscanl\<cdot>f'\<cdot>(unwrap\<cdot>z)"
Fax_def: shows "Fax \<equiv> {fax f |f. wtFsym f \<and> \<not> unprot (resOf f) \<and> \<not> protCl (resOf f)} \<union> {faxCD f |f. wtFsym f \<and> protCl (resOf f)}"
sets_density: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> ennreal" shows "sets (density M f) = sets M"
simps: fixes a :: "'c llist" shows "ldistinct a = (a = LNil \<or> (\<exists>x xs. a = LCons x xs \<and> x \<notin> lset xs \<and> ldistinct xs))"
Confidentiality_A_temporal: fixes A :: "agent"   and T :: "nat"   and B :: "agent"   and K :: "nat"   and X :: "msg"   and evs :: "event list" assumes "Crypt (shrK A) \<lbrace>Number T, Agent B, Key K, X\<rbrace> \<in> parts (knows Spy evs)"   and "\<not> expiredK T evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> bankerberos" shows "Key K \<notin> analz (knows Spy evs)"
set_poly: shows "set (poly_mult p q) \<subseteq> carrier R"
remdups_mset_empty: shows "remdups_mset {#} = {#}"
abs_eq: fixes xa :: "nat \<Rightarrow>\<^sub>0 nat"   and x :: "'b" shows "MPoly_Type.monom xa x = MPoly (Poly_Mapping.single xa x)"
eq_OO: fixes R :: "'n \<Rightarrow> 'o \<Rightarrow> bool" shows "(=) OO R = R"
Let_def: fixes s :: "'f"   and f :: "'f \<Rightarrow> 'e" shows "Let s f \<equiv> f s"
simps_2: fixes f :: "'b \<Rightarrow> bool"   and a :: "'b"   and as :: "'b negation_type list" shows "cnf_to_bool f (Pos a # as) = (f a \<and> cnf_to_bool f as)"
match_rews_6: fixes a :: "'r"   and tree1 :: "'r tree"   and tree2 :: "'r tree"   and k :: "'r \<rightarrow> 'r tree \<rightarrow> 'r tree \<rightarrow> 's match" assumes "a \<noteq> \<bottom>" shows "match_Node\<cdot>(Node\<cdot>a\<cdot>tree1\<cdot>tree2)\<cdot>k = k\<cdot>a\<cdot>tree1\<cdot>tree2"
tranclp_trans_induct: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "r\<^sup>+\<^sup>+ x y"   and "\<And>x y. r x y \<Longrightarrow> P x y"   and "\<And>x y z. \<lbrakk>r\<^sup>+\<^sup>+ x y; P x y; r\<^sup>+\<^sup>+ y z; P y z\<rbrakk> \<Longrightarrow> P x z" shows "P x y"
cont_cfg_corec: fixes t :: "'a"   and d :: "'b \<Rightarrow> 'a pmf"   and x :: "'b"   and s :: "'a"   and c :: "'b \<Rightarrow> 'a \<Rightarrow> 'b" assumes "t \<in> set_pmf (d x)" shows "cont (cfg_corec s d c x) t = cfg_corec t d c (c x t)"
h1: fixes x :: "'c" shows "h (1, x) = f' x"
cseq: shows "cs\<^bsup>path \<sigma>\<^esup> k = cs\<^bsup>path \<sigma>'\<^esup> k'"
heap_ordered_single: fixes t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" shows "SkewBinomialHeapStruc.heap_ordered t = (\<forall>x\<in>#SkewBinomialHeapStruc.tree_to_multiset t. SkewBinomialHeapStruc.prio t \<le> snd x)"
LVar: fixes G :: "prog"   and s :: "st"   and vn :: "lname"   and n :: "nat" shows "G\<turnstile>Norm s \<midarrow>LVar vn=\<succ>lvar vn s\<midarrow>n\<rightarrow> Norm s"
fps_monom_aux: fixes k :: "nat" shows "Abs_fps (\<lambda>n. of_nat n ^ k) = fps_of_poly (fps_monom_poly (1::'b) k) / (1 - fps_X) ^ (k + 1)"
neighbourhood_extension_into_ANR: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and T :: "'b set" assumes "continuous_on S f"   and "f ` S \<subseteq> T"   and "ANR T"   and "closed S" obtains V :: "'a set" and g :: "'a \<Rightarrow> 'b" where "S \<subseteq> V" and "open V" and "continuous_on V g" and "g ` V \<subseteq> T" and "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
functor_abbrevE: fixes F :: "('d, 'e, 'f, 'g, 'h, 'i, 'j) Functor_scheme"   and A :: "('d, 'f, 'h) Category_scheme"   and B :: "('e, 'g, 'i) Category_scheme" assumes "Ftor F : A \<longrightarrow> B" obtains "Functor F" and "CatDom F = A" and "CatCod F = B"
suff_finpref: fixes t :: "'b llist"   and A :: "'b set"   and r :: "'b llist" assumes "t \<in> suff A r"   and "r \<in> A\<^sup>\<star>" shows "r \<in> finpref A t"
numer_cf5_def: shows "numer_cf5 \<equiv> \<lambda>x. x ^ 5 + 30 * x ^ 4 + 420 * x ^ 3 + 3360 * x\<^sup>2 + 15120 * x + 30240"
Procedure_defs_4: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> ('a, 'b) chan"   and p :: "PrIds"   and v :: "state \<Rightarrow> 'a" shows "Call ch p v \<equiv> \<not> $Calling ch p \<and> cbit<ch!p>$ \<noteq> $rbit<ch!p> \<and> arg<ch!p>$ = $v"
simps_5: fixes f :: "'f"   and b :: "'d set"   and c\<^sub>1 :: "('d, 'e, 'g) com"   and c\<^sub>2 :: "('d, 'e, 'g) com" shows "mark_guards f (Cond b c\<^sub>1 c\<^sub>2) = Cond b (mark_guards f c\<^sub>1) (mark_guards f c\<^sub>2)"
redT_updT_new_thread: fixes ts :: "'h \<Rightarrow> ('f \<times> 'g \<Rightarrow>f nat) option"   and ta :: "('h, 'f, 'i) new_thread_action"   and t :: "'h"   and x :: "'f"   and w :: "'g \<Rightarrow>f nat" assumes "redT_updT ts ta t = \<lfloor>(x, w)\<rfloor>"   and "thread_ok ts ta"   and "ts t = None" shows "\<exists>m. ta = NewThread t x m \<and> w = no_wait_locks"
est: fixes x :: "real"   and y :: "real" assumes "x \<le> 1"   and "0 \<le> y" shows "x * y \<le> 1 * y"
convex_abs_powr: fixes p :: "real" assumes "1 \<le> p" shows "convex_on UNIV (\<lambda>x. \<bar>x\<bar> powr p)"
min_dist_succ: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and v :: "nat"   and v' :: "nat"   and v'' :: "nat" assumes "Graph.connected c v v'"   and "(v', v'') \<in> Graph.E c" shows "Graph.min_dist c v v'' \<le> Suc (Graph.min_dist c v v')"
poly_reflect_poly_nz: fixes x :: "'a"   and p :: "'a poly" assumes "x \<noteq> (0::'a)" shows "poly (reflect_poly p) x = x ^ degree p * poly p (inverse x)"
maxim_in: fixes A :: "'a set" assumes "finite A"   and "A \<noteq> {}"   and "A \<subseteq> Field r" shows "maxim A \<in> A"
unit_in_hom_2: fixes a :: "'c" assumes "C.obj a" shows "\<guillemotleft>H.unit a : H.map\<^sub>0 a \<Rightarrow>\<^sub>D H a\<guillemotright>"
sort_of_ty_eq_iff: fixes x :: "ty"   and y :: "ty" shows "(sort_of_ty x = sort_of_ty y) = (x = y)"
setdiff_irrelevant: fixes A :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "finite A" shows "prod g (A - {x. g x = (1::'c)}) = prod g A"
SN_ars_axioms: shows "SN_ars (s_mul_ext NS S)"
inj_map: fixes f1 :: "'c \<Rightarrow> 'd"   and f2 :: "'e \<Rightarrow> 'f" assumes "inj f1"   and "inj f2" shows "inj (map_prod f1 f2)"
BPd'_def: shows "BPd' = BPd(Xd d2h1 := {})"
Domainp_sI: fixes x :: "'a"   and y :: "'c" assumes "Domainp A x" shows "Domainp A (s y x)"
AE_PiM_exponential_suminf_infty: fixes R :: "nat \<Rightarrow> real" assumes "\<And>n. 0 < R n"   and "(\<Sum>n. ennreal (1 / R n)) = \<top>" shows "AE \<omega> in Pi\<^sub>M UNIV (\<lambda>n. exponential (R n)). (\<Sum>n. ereal (\<omega> n)) = \<infinity>"
VPow_vsubset_Vset: fixes X :: "V"   and i :: "V" assumes "X \<in>\<^sub>\<circ> Vset i" shows "VPow X \<subseteq>\<^sub>\<circ> Vset i"
simps_7: fixes \<phi> :: "atom fm" shows "nnf (AllQ \<phi>) = AllQ (nnf \<phi>)"
lappend_llist_of_LCons: fixes xs :: "'c list"   and y :: "'c"   and ys :: "'c llist" shows "lappend (llist_of xs) (LCons y ys) = lappend (llist_of (xs @ [y])) ys"
remdups_eq_nil_iff: fixes x :: "'e list" shows "(remdups x = []) = (x = [])"
sort_key_on_simps_1: fixes f :: "'b \<Rightarrow> 'a" shows "sort_key_on r f [] = []"
converse_tranclpE: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and z :: "'b" assumes "r\<^sup>+\<^sup>+ x z" obtains "r x z"   | y :: "'b" where "r x y" and "r\<^sup>+\<^sup>+ y z"
sigma_algebra_pre_sigma: fixes \<Omega> :: "'c set"   and F :: "'d \<Rightarrow> 'c measure"   and T :: "'c \<Rightarrow> 'd" assumes "filtration \<Omega> F"   and "stopping_time F T" shows "sigma_algebra \<Omega> {A. \<forall>t. {\<omega> \<in> A. T \<omega> \<le> t} \<in> sets (F t)}"
simps_2: fixes p :: "nat"   and a :: "floatarith"   and b :: "floatarith"   and vs :: "((real \<times> real pdevs) \<times> real) list" shows "approx_floatarith p (Mult a b) vs = approx_floatarith p a vs \<bind> (\<lambda>a1. approx_floatarith p b vs \<bind> (\<lambda>a2. Some (mult_aform' p a1 a2)))"
wellformed: fixes b :: "'a set"   and bl :: "'a set" assumes "b \<in># add_point_sys.add_block bl" shows "b \<subseteq> \<V> \<union> bl"
subset_class_axioms: shows "subset_class \<Omega> (Dynkin \<Omega> M)"
exec_def: fixes n :: "nat"   and xs :: "nat list" shows "exec n xs = map (\<lambda>pc. (pc, transf n xs)) (rev (sorted_list_of_set (succs n)))"
idiff_infinity: fixes n :: "eint" shows "\<infinity> - n = \<infinity>"
absorb_iff2: fixes b :: "'a upred"   and a :: "'a upred" shows "(a \<sqsubseteq> b) = ((a \<and> b) = b)"
ivl_integral_has_vderiv_on_compact_interval: fixes A :: "real set"   and f :: "real \<Rightarrow> 'a"   and c :: "real" assumes "continuous_on A f"   and "c \<in> A"   and "is_interval A"   and "compact A" shows "((\<lambda>u. ivl_integral c u f) has_vderiv_on f) A"
semigroup_ow_axioms: shows "semigroup_ow U (\<^bold>*\<^sub>o\<^sub>w)"
orderP: shows "order (G\<lparr>carrier := P\<rparr>) = p ^ a * k"
Pi_I': fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'f"   and B :: "'e \<Rightarrow> 'f set" assumes "\<And>x. x \<in> A \<longrightarrow> f x \<in> B x" shows "f \<in> Pi A B"
reflect_poly_hom: fixes hom :: "'a \<Rightarrow> 'b"   and p :: "'a poly" assumes "inj_idom_hom hom" shows "reflect_poly (map_poly hom p) = map_poly hom (reflect_poly p)"
config'_snoc: fixes step :: "'b \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'b"   and A :: "('b \<Rightarrow> 'c) \<times> ('b \<times> 'c \<Rightarrow> 'd \<Rightarrow> 'e \<times> 'c)"   and s :: "'b \<times> 'c"   and rs :: "'d list"   and r :: "'d" shows "On_Off.config' step A s (rs @ [r]) = On_Off.Step step A (On_Off.config' step A s rs) r"
not_Some_eq: fixes x :: "'g option" shows "(\<forall>y. x \<noteq> Some y) = (x = None)"
chain_lmap: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and R' :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and xs :: "'a llist" assumes "\<forall>x y. R x y \<longrightarrow> R' (f x) (f y)"   and "chain R xs" shows "chain R' (lmap f xs)"
ord_eq_le_trans: fixes a :: "'c"   and b :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "'c" assumes "a = b"   and "less_eq b c" shows "less_eq a c"
wca_onI: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "\<And>B x y. \<lbrakk>B \<subseteq> A; (x, y) \<in> rwp_on A f; x \<in> f B; y \<in> B\<rbrakk> \<Longrightarrow> y \<in> f B" shows "wca_on A f"
decode_instr_result_1: fixes v1 :: "word32" assumes "\<nexists>v2. decode_instruction v1 = Inr v2" shows "\<exists>e. decode_instruction v1 = Inl e"
assms_4: shows "Separation (remove_vertex v) v0 v1 S"
lgrid_empty': fixes lm :: "nat"   and p :: "(nat \<times> int) list"   and ds :: "nat set" assumes "lm \<le> level p" shows "lgrid p ds lm = {}"
oreachable_subterms: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp"   and A :: "((nat \<Rightarrow> 'f) \<times> ('f, 'g, 'e, 'h) seqp, 'g seq_action) automaton"   and i :: "nat"   and \<sigma> :: "nat \<Rightarrow> 'f"   and p :: "('f, 'g, 'e, 'h) seqp"   and S :: "(nat \<Rightarrow> 'f) \<Rightarrow> (nat \<Rightarrow> 'f) \<Rightarrow> 'g seq_action \<Rightarrow> bool"   and U :: "(nat \<Rightarrow> 'f) \<Rightarrow> (nat \<Rightarrow> 'f) \<Rightarrow> bool" assumes "wellformed \<Gamma>"   and "control_within \<Gamma> (init A)"   and "automaton.trans A = oseqp_sos \<Gamma> i"   and "(\<sigma>, p) \<in> oreachable A S U" shows "\<exists>pn. p \<in> subterms (\<Gamma> pn)"
eqvt_at_def: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b" shows "eqvt_at f x \<equiv> \<forall>p. p \<bullet> f x = f (p \<bullet> x)"
disj_matrices_contr1: fixes A :: "'a matrix"   and B :: "'a matrix"   and j :: "nat"   and i :: "nat" assumes "disj_matrices A B"   and "Rep_matrix A j i \<noteq> (0::'a)" shows "Rep_matrix B j i = (0::'a)"
newpkt_def: fixes p :: "('i, 'pkt_ext) tagged_packet_scheme" shows "newpkt p \<equiv> match_tcp_flags ipt_tcp_syn (p_tcp_flags p) \<and> p_tag_ctstate p = CT_New"
fresh_singleton: fixes a :: "'a"   and x :: "'b" shows "a \<sharp> {x} = a \<sharp> x"
simps_7: fixes SG :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and C :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and i :: "nat"   and j :: "nat"   and k :: "nat" shows "step_result (SG, C) (i, CloseId j k) = (closeI SG i, C)"
assms_1: shows "X \<in> obj\<^bsub>A\<^esub>"
closest_supchamber_closest: fixes C :: "'a set"   and F :: "'a set"   and D :: "'a set" assumes "chamber C"   and "order.greater_eq C F" shows "order.greater_eq (chamber_distance C D) (chamber_distance (closest_supchamber F D) D)"
Bossy_CH_range: fixes X :: "'a set"   and X' :: "'a set" assumes "X \<subseteq> Bossy'.CH X'" shows "X \<subseteq> set [d1h1, d1h3, d2h1, d2h3, d3h1, d3h3]"
chain: shows "Complete_Partial_Order.chain (\<le>) (A \<union> B)"
mset_distrib: fixes A :: "'f multiset"   and B :: "'f multiset"   and M :: "'f multiset"   and N :: "'f multiset" assumes "A + B = M + N" obtains Am :: "'f multiset" and An :: "'f multiset" and Bm :: "'f multiset" and Bn :: "'f multiset" where "A = Am + An" and "B = Bm + Bn" and "M = Am + Bm" and "N = An + Bn"
2: shows "bprv (imp (PP \<langle>\<phi>G\<rangle>) (PP \<langle>neg (PP \<langle>\<phi>G\<rangle>)\<rangle>))"
fs_carr: fixes j :: "nat" assumes "j < length fs" shows "fs ! j \<in> carrier_vec n"
CR_onD: fixes r :: "('a \<times> 'a) set"   and A :: "'a set"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "CR_on r A"   and "a \<in> A"   and "(a, b) \<in> r\<^sup>*"   and "(a, c) \<in> r\<^sup>*" shows "(b, c) \<in> r\<^sup>\<down>"
word_eq_iff_signed: fixes one :: "'b"   and times :: "'b \<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 v :: "'c word"   and w :: "'c word" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(v = w) = (ring_1.signed one plus zero minus v = ring_1.signed one plus zero minus w)"
norm_beta_complete: fixes A :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and l :: "'d"   and D :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry"   and l' :: "'d"   and Z :: "('c \<Rightarrow> real) set" assumes "A \<turnstile> \<langle>l, [D]\<^bsub>v,n\<^esub>\<rangle> \<leadsto>\<^sub>\<beta> \<langle>l', Z\<rangle>"   and "(\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)) \<and> (\<forall>c\<in>clk_set A. v c \<le> n) \<and> (\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k))"   and "valid_abstraction A X (\<lambda>x. real (k x))"   and "valid_dbm D" obtains D' :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" where "A \<turnstile> \<langle>l, D\<rangle> \<leadsto>\<^sub>\<N> \<langle>l', D'\<rangle>" and "[D']\<^bsub>v,n\<^esub> = Z" and "valid_dbm D'"
4: fixes f :: "complex \<Rightarrow> complex" assumes "f holomorphic_on h ` S"   and "\<And>z. z \<in> h ` S \<Longrightarrow> f z \<noteq> 0"   and "inj_on f (h ` S)" shows "\<exists>g. g holomorphic_on h ` S \<and> (\<forall>z\<in>h ` S. f z = (g z)\<^sup>2)"
less_eq_flat_complete_lattice_def: fixes x :: "'b flat_complete_lattice"   and y :: "'b flat_complete_lattice" shows "x \<le> y \<equiv> case x of flat_complete_lattice.Value v1 \<Rightarrow> case y of flat_complete_lattice.Value v2 \<Rightarrow> v1 = v2 | flat_complete_lattice.Bot \<Rightarrow> False | flat_complete_lattice.Top \<Rightarrow> True | flat_complete_lattice.Bot \<Rightarrow> True | flat_complete_lattice.Top \<Rightarrow> y = flat_complete_lattice.Top"
simps_2: fixes x :: "'d"   and f :: "'d \<Rightarrow> 'c option" shows "Some x \<bind> f = f x"