hcju/isabelleps huggingface dataset:
valid_slide: fixes w :: "nat \<times> nat"   and t :: "'a tree" assumes "0 < fst w"   and "fst w \<le> snd w"   and "local.valid t"   and "l t \<le> fst w"   and "r t \<le> snd w"   and "snd w \<le> length as" shows "local.valid (local.slide t w) \<and> l (local.slide t w) = fst w \<and> r (local.slide t w) = snd w"
idem1: fixes x :: "'a"   and y :: "'a" shows "c (b x y) x"
function_ring_is_monoid: shows "Group.monoid (function_ring (carrier Zp) Zp)"
cMax_finite_ex: fixes X :: "'b set" assumes "finite X"   and "X \<noteq> {}" shows "\<exists>x\<in>X. Max X = x"
leg0_in_hom_1: shows "\<guillemotleft>p\<^sub>0 : src p\<^sub>0 \<rightarrow> src (k\<^sup>* \<star> h)\<guillemotright>"
vdw_2_right: fixes k :: "nat" shows "vdw k 2 (k + 1)"
iso_dghm_ArrMap_v11: shows "v11 (\<FF>\<lparr>ArrMap\<rparr>)"
FinGroupRepresentation_FG: shows "FinGroupRepresentation G (*) FG"
lr_of_tran_correct_1: fixes rt :: "32 routing_rule list"   and fw :: "32 simple_rule list"   and ifs :: "char list list"   and oft :: "(of_match_field, of_action list) flow_entry_match list"   and p :: "(32, 'b) simple_packet_ext_scheme"   and oif :: "char list" assumes "lr_of_tran rt fw ifs = Inr oft"   and "p_l2type p = 2048"   and "p_iiface p \<in> set ifs" shows "(OF_priority_match OF_match_fields_safe oft p = Action [Forward oif]) = (simple_linux_router_nol12 rt fw p = Some (p\<lparr>p_oiface := oif\<rparr>))"
lang_norm: fixes r :: "'a rexp" shows "lang (norm r) = lang r"
that: fixes z :: "'a" assumes "r\<^sup>*\<^sup>* y1 z"   and "r\<^sup>*\<^sup>* y2 z" shows "thesis"
fls_shift_nonneg_subdegree: fixes m :: "int"   and f :: "'a fls" assumes "m \<le> fls_subdegree f" shows "0 \<le> fls_subdegree (fls_shift m f)"
IH: fixes x :: "'a vec" assumes "set Vs \<subseteq> set Ws"   and "x \<in> convex_hull_list Vs" shows "x \<in> convex_hull_list Ws"
le_gfp: shows "(\<lambda>xs. alw (HLD Y) xs \<and> alw (\<lambda>\<omega>. HLD Y \<omega> \<longrightarrow> ev (HLD X) \<omega>) xs) \<le> gfp run"
hassoc_map_assn_hassoc_delete: fixes m :: "'a \<Rightarrow> 'b option"   and p :: "('a \<times> 'b) list"   and k :: "'a" shows "<hassoc_map_assn m p> hassoc_delete k p <hassoc_map_assn (m |` (- {k}))>\<^sub>t"
h: shows "h \<in> N \<rightarrow>\<^sub>M M i"
eats_cong_ax_holds: fixes e :: "(name, hf) finfun" shows "eval_fm e eats_cong_ax"
list_of_eucl_def: fixes x :: "'b" shows "list_of_eucl x = map ((\<bullet>) x) Basis_list"
im_diff: fixes g :: "'g"   and g' :: "'g" shows "T (g - g') = T g - T g'"
try_bind_spmf_lossless2: fixes q :: "'c spmf"   and p :: "'d spmf"   and f :: "'d \<Rightarrow> 'c spmf" assumes "lossless_spmf q" shows "TRY p \<bind> f ELSE q = TRY p \<bind> (\<lambda>x. TRY f x ELSE q) ELSE q"
prems: fixes m :: "nat" assumes "P m" shows "m < Suc n"
delay_bisimulation_diverge_flip: shows "delay_bisimulation_diverge trsys2 trsys1 (flip bisim) (flip tlsim) \<tau>move2 \<tau>move1"
fin_1: shows "dim (span ((+) (- a) ` (B - {a}))) = card ((+) (- a) ` (B - {a}))"
simps_7: fixes x3 :: "agent"   and x11 :: "agent"   and x12 :: "agent" shows "privK x3 \<noteq> sharK x11 x12"
catch_def: fixes G :: "prog"   and s :: "state"   and C :: "qtname" shows "G,s\<turnstile>catch C = (\<exists>xc. fst s = Some (Xcpt xc) \<and> G,snd s\<turnstile>Addr (the_Loc xc) fits Class C)"
resource_of_obsf_oracle_Fault: fixes "oracle" :: "'c \<Rightarrow> 'a \<Rightarrow> ('b \<times> 'c) spmf" shows "RES (obsf_oracle oracle) Fault = const_resource Fault"
induct: fixes x :: "'b set"   and A :: "'b set set"   and P :: "'b set \<Rightarrow> bool" assumes "x \<in> sigma A"   and "\<And>a. a \<in> A \<Longrightarrow> P a"   and "P {}"   and "\<And>a. \<lbrakk>a \<in> sigma A; P a\<rbrakk> \<Longrightarrow> P (- a)"   and "\<And>a. \<lbrakk>\<And>i. a i \<in> sigma A; \<And>i. P (a i)\<rbrakk> \<Longrightarrow> P (\<Union> (range a))" shows "P x"
delete_only_m_effect: fixes x :: "('a, 'b) insert_message" shows "delete_only_m (to_woot_character M x) = to_woot_character (M \<union> {Delete m}) x"
mismatchNil: fixes a :: "name"   and P :: "pi" shows "[a\<noteq>a]P \<simeq> \<zero>"
facet_of_convex_hull_affine_independent: fixes S :: "'b set"   and T :: "'b set" assumes "\<not> affine_dependent S" shows "(T facet_of convex hull S) = (T \<noteq> {} \<and> (\<exists>u. u \<in> S \<and> T = convex hull (S - {u})))"
HDomain_Incl_Subset: fixes H :: "fm set"   and r :: "tm"   and k :: "tm"   and k' :: "tm" assumes "H \<turnstile> HDomain_Incl r k"   and "H \<turnstile> k' SUBS k" shows "H \<turnstile> HDomain_Incl r k'"
nrename_OrL_aux: fixes R :: "trm"   and x :: "name"   and y :: "name"   and v :: "name"   and M :: "trm"   and w :: "name"   and N :: "trm"   and u :: "name" assumes "R[x\<turnstile>n>y] = OrL v.M w.N u" shows "x = u \<and> x = y \<or> x \<noteq> u"
grid_base_base: fixes b :: "(nat \<times> int) list"   and dm :: "nat"   and ds' :: "nat set"   and ds :: "nat set" assumes "b \<in> sparsegrid' dm" shows "base ds' b \<in> grid (base ds (base ds' b)) (ds \<union> ds')"
stack_ss_tj_stack: fixes G :: "('c, 'd) graph_rec_scheme"   and s :: "'c tarjan_state" assumes "Tarjan_invar G s" shows "set (stack s) \<subseteq> set (tj_stack s)"
det_dim_zero: fixes A :: "'a mat" assumes "A \<in> carrier_mat 0 0" shows "det A = (1::'a)"
unit_imp_dvd: 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 b :: "'b"   and a :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times b one" shows "dvd.dvd times b a"
routing_ipassmt_def: fixes rt :: "'a routing_rule list" shows "routing_ipassmt rt \<equiv> map ((apfst Iface \<circ>\<circ> apsnd) cidr_split) (routing_ipassmt_wi rt)"
abelian_groupE_4: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b"   and y :: "'b" assumes "abelian_group R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y = y \<oplus>\<^bsub>R\<^esub> x"
Exe1I: fixes F :: "\<Pi>\<^sub>1"   and x :: "\<kappa>"   and v :: "i" assumes "\<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> ex1 r v" shows "[\<lparr>F,x\<rparr> in v]"
trapped_iff_on_existence_ivl0: fixes x :: "'a"   and K :: "'a set" shows "rev.trapped x K = (rev.flow0 x ` rev.existence_ivl0 x \<subseteq> K)"
name_fresh_fresh: fixes a :: "name"   and x :: "'a"   and b :: "name" assumes "a \<sharp> x"   and "b \<sharp> x" shows "[(a, b)] \<bullet> x = x"
Cons_nth_drop_Suc: fixes i :: "nat"   and xs :: "'g list" assumes "i < length xs" shows "xs ! i # drop (Suc i) xs = drop i xs"
cone_cone_hull: fixes S :: "'c set" shows "cone (cone hull S)"
intro: fixes U :: "'a set"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ls :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a" assumes "lattice_ow U inf le ls sup"   and "order_bot_ow U bot le ls" shows "bounded_lattice_bot_ow U inf le ls sup bot"
theFT_finished: fixes n :: "nat"   and cl :: "('a, 'b, 'c) cmd list" assumes "n \<in> theFT cl" shows "finished (cl ! n)"
simps_4: fixes n :: "nat"   and f :: "recf"   and gs :: "recf list" shows "arity (Cn n f gs) = n"
ord_le_eq_trans: fixes a :: "'c list"   and b :: "'c list"   and c :: "'c list" assumes "suffix a b"   and "b = c" shows "suffix a c"
analytic_at_ball: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex" shows "(f analytic_on {z}) = (\<exists>e>0. f holomorphic_on ball z e)"
case_eq_if: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and prod :: "'b \<times> 'c" shows "(case prod of (x, xa) \<Rightarrow> f x xa) = f (cc prod) (snd prod)"
term_ok'_occs: fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and t :: "Core.term"   and u :: "Core.term" assumes "is_std_sig \<Sigma>"   and "term_ok' \<Sigma> t"   and "occs u t" shows "term_ok' \<Sigma> u"
upd_conv_take_nth_drop: fixes i :: "nat"   and xs :: "'b list"   and a :: "'b" assumes "i < length xs" shows "xs[i := a] = take i xs @ a # drop (Suc i) xs"
hyps: shows "of_real_hom.mat_hom A ^\<^sub>m k * P = P * J ^\<^sub>m k"
MM_axioms_def: shows "MM_axioms \<equiv> \<forall>y x. PP y x \<longrightarrow> (\<exists>z. P z x \<and> \<not> O z y)"
erlang_distributed_le: fixes X :: "'a \<Rightarrow> real"   and k :: "nat"   and l :: "real"   and a :: "real" assumes "distributed M lborel X (\<lambda>x. ennreal (erlang_density k l x))"   and "0 < l"   and "0 \<le> a" shows "prob {x \<in> space M. X x \<le> a} = erlang_CDF k l a"
awalk_app_iso_eq: fixes hom :: "('c, 'd, 'e, 'f) digraph_isomorphism"   and u :: "'e"   and v :: "'e"   and p :: "'f list" assumes "H.digraph_isomorphism hom"   and "u \<in> iso_verts hom ` verts (app_iso hom G)"   and "v \<in> iso_verts hom ` verts (app_iso hom G)"   and "set p \<subseteq> iso_arcs hom ` arcs (app_iso hom G)" shows "pre_digraph.awalk (app_iso hom (app_iso hom G)) u p v = H.awalk (iso_verts (H.inv_iso hom) u) (map (iso_arcs (H.inv_iso hom)) p) (iso_verts (H.inv_iso hom) v)"
infl: shows "\<forall>x\<in>A. \<forall>y\<in>A. x \<sqsubseteq> y \<longrightarrow> x \<sqsubseteq> f y"
inverse_le_1_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and 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 less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq (inverse x) one = (less_eq x zero \<or> less_eq one x)"
homeomorphic_maps_def: fixes X :: "'e topology"   and Y :: "'f topology"   and f :: "'e \<Rightarrow> 'f"   and g :: "'f \<Rightarrow> 'e" shows "homeomorphic_maps X Y f g \<equiv> continuous_map X Y f \<and> continuous_map Y X g \<and> (\<forall>x\<in>topspace X. g (f x) = x) \<and> (\<forall>y\<in>topspace Y. f (g y) = y)"
ceiling_add_one: fixes x :: "'a" shows "\<lceil>x + (1::'a)\<rceil> = \<lceil>x\<rceil> + 1"
gt_sub_arg: fixes s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" assumes "wary (App s t)" shows "App s t >\<^sub>t t"
simps_8: fixes v :: "poly"   and va :: "nat" shows "degreen (Pw v va) = (\<lambda>m. 0)"
xxx: shows "Re (A - D - cor dsc) / Re (2 * (B * cnj B)) \<le> Re (A - D + cor dsc) / Re (2 * (B * cnj B))"
assms_3: fixes x :: "'a acom" assumes "P x" shows "P (f x)"
lm116: fixes ll :: "'b"   and l :: "'b list" assumes "ll \<in> set l" shows "\<exists>n. ll = l ! n \<and> n < length l \<and> 0 \<le> n"
inverse_unique: fixes f :: "'b"   and g :: "'b" assumes "inverse_arrows f g" shows "local.inv f = g"
DirProds_empty: fixes f :: "'c \<Rightarrow> ('d, 'e) monoid_scheme" shows "carrier (DirProds f {}) = {\<one>\<^bsub>DirProds f {}\<^esub>}"
sum_list_nonpos: fixes xs :: "'a multiset list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> x \<subseteq># {#}" shows "monoid_add.sum_list (+) {#} xs \<subseteq># {#}"
assms_2: shows "fps_nth F 0 \<noteq> (0::'a)"
assms_2: shows "\<forall>r\<in>R. Well_order r"
K_imply_weaken: fixes ps :: "'b fm list"   and q :: "'b fm"   and ps' :: "'b fm list" assumes "\<turnstile> imply ps q"   and "set ps \<subseteq> set ps'" shows "\<turnstile> imply ps' q"
SINK_mono: fixes f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "\<And>e. f e \<le> g e" shows "SINK g \<subseteq> SINK f"
Eats_cong: fixes H :: "fm set"   and t :: "tm"   and t' :: "tm"   and u :: "tm"   and u' :: "tm" assumes "H \<turnstile> t EQ t'"   and "H \<turnstile> u EQ u'" shows "H \<turnstile> Eats t u EQ Eats t' u'"
seqE: fixes g :: "(bool, 'arr list) concrete_category.arr"   and f :: "(bool, 'arr list) concrete_category.arr" assumes "seq g f" obtains "arr f" and "arr g" and "local.dom g = cod f"
of_int_of_nat: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and k :: "int" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus k = (if k < 0 then uminus (semiring_1.of_nat one plus zero (nat (- k))) else semiring_1.of_nat one plus zero (nat k))"
conf_clinit_Cons_nclinit: fixes P :: "jvm_prog"   and sh :: "sheap"   and frs :: "frame list"   and M :: "char list"   and stk :: "val list"   and loc :: "val list"   and C :: "char list"   and pc :: "nat" assumes "conf_clinit P sh frs"   and "M \<noteq> clinit" shows "conf_clinit P sh ((stk, loc, C, M, pc, No_ics) # frs)"
disjnt_insert2: fixes Y :: "'a set"   and a :: "'a"   and X :: "'a set" shows "disjnt Y (insert a X) = (a \<notin> Y \<and> disjnt Y X)"
Range_outside_sub: fixes R :: "('d \<times> 'c) set"   and Y :: "'c set"   and X :: "'d set" assumes "Range R \<subseteq> Y" shows "Range (Outside' X R) \<subseteq> Y"
two_block_non_interfering_dims_1: fixes A :: "'a mat"   and B :: "'a mat" shows "dim_row (two_block_non_interfering A B) = dim_row A + dim_row B"
Cons_mem_set_subseqs_append: fixes x :: "'c"   and ys :: "'c list"   and xs :: "'c list"   and zs :: "'c list" assumes "x \<in> set ys"   and "xs \<in> set (subseqs zs)" shows "x # xs \<in> set (subseqs (ys @ zs))"
flipped_reflections_odd_lconjseq: fixes ss :: "'w list" assumes "ss \<in> lists S" shows "flipped_reflections (sum_list ss) = {t \<in> \<Union>w\<in>W. lconjby w ` S. \<not> gcd_nat.greater_eq (count_list (lconjseq ss) t) 2}"
get_elements_by_class_name_impl: shows "get_elements_by_class_name = local.a_get_elements_by_class_name"
sel_4: fixes x1 :: "'b"   and x2 :: "'b"   and x3 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x4 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x5 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x6 :: "'b \<Rightarrow> 'b"   and x7 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x8 :: "'b \<Rightarrow> 'b"   and x9 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x10 :: "'b \<Rightarrow> 'b"   and x11 :: "'b \<Rightarrow> 'b"   and x12 :: "int \<Rightarrow> 'b"   and x13 :: "'b \<Rightarrow> int"   and x14 :: "'b \<Rightarrow> bool" shows "arith_ops_record.times (Arith_Ops_Record x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14) = x4"
encL: fixes S :: "'procS"   and SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" shows "TargetTerm (\<lbrakk>S\<rbrakk>) \<lesssim>\<lbrakk>\<cdot>\<rbrakk>L<SRel,TRel> SourceTerm S"
adverse_in_R3: fixes z :: "nat" shows "\<exists>r. recfn 3 r \<and> Partial_Recursive.total r \<and> (\<lambda>i j x. eval r [i, j, x]) = local.adverse z"
Cast_final: fixes C :: "char list"   and e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option" assumes "P,E \<turnstile> \<langle>\<lparr>C\<rparr>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>"   and "final e'" shows "\<exists>e'' s''. P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e'',s''\<rangle> \<and> P,E \<turnstile> \<langle>\<lparr>C\<rparr>e'',s''\<rangle> \<rightarrow> \<langle>e',s'\<rangle> \<and> final e''"
alive_access: fixes s :: "Store"   and l :: "Location" shows "alive (s@@l) s"
assms: fixes N :: "nat" shows "of_nat N \<le> x"
degree_mult_eq: fixes p :: "'b poly"   and q :: "'b poly" assumes "p \<noteq> 0"   and "q \<noteq> 0" shows "degree (p * q) = degree p + degree q"
p_def: shows "p = Abs_poly (\<lambda>k. if k \<le> n then f k else (0::'a))"
rquot_D: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<preceq>\<^sub>R y"   and "z = rquot y x" shows "D x z"
simps_2: fixes literal :: "Literal"   and list :: "Literal list" shows "varsClause (literal # list) = {var literal} \<union> varsClause list"
InitBlockReds: fixes e :: "expr"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and V :: "char list"   and v' :: "val"   and e' :: "expr"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l' :: "char list \<Rightarrow> val option"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and T :: "ty"   and v :: "val"   and wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "P,E(V \<mapsto> T) \<turnstile> \<langle>e,(h, l(V \<mapsto> v'))\<rangle> \<rightarrow>* \<langle>e',(h', l')\<rangle>"   and "P \<turnstile> T casts v to v' "   and "wf_prog wf_md P" shows "\<exists>v'' w. P,E \<turnstile> \<langle>{V:T; V:=Val v;; e},(h, l)\<rangle> \<rightarrow>* \<langle>{V:T; V:=Val v'';; e'},(h', l'(V := l V))\<rangle> \<and> P \<turnstile> T casts v'' to w "
round_interval_mono: fixes X :: "float interval"   and Y :: "float interval"   and prec :: "nat" assumes "set_of X \<subseteq> set_of Y" shows "set_of (round_interval prec X) \<subseteq> set_of (round_interval prec Y)"
M_plus_1: fixes m :: "int"   and x :: "int"   and y :: "int" shows "poly_mod.M m (poly_mod.M m x + y) = poly_mod.M m (x + y)"
mult_cancel_right1: fixes c :: "'a"   and b :: "'a" shows "(c = b * c) = (c = (0::'a) \<or> b = (1::'a))"
leQ: shows "Q \<le>\<^sub>L 1\<^sub>m d"
divide_divide_times_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and 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 x :: "'a"   and y :: "'a"   and z :: "'a"   and w :: "'a" assumes "class.field divide inverse times one plus zero minus uminus" shows "divide (divide x y) (divide z w) = divide (times x w) (times y z)"
span_mono: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B" shows "span A \<subseteq> span B"
continuous_on_o_Pair: fixes T :: "'c set"   and X :: "'d set"   and h :: "'c \<times> 'd \<Rightarrow> 'e"   and t :: "'c" assumes "continuous_on (T \<times> X) h"   and "t \<in> T" shows "continuous_on X (h \<circ> Pair t)"
rtrancl_into_rtrancl: fixes a :: "'e"   and b :: "'e"   and r :: "('e \<times> 'e) set"   and c :: "'e" assumes "(a, b) \<in> r\<^sup>*"   and "(b, c) \<in> r" shows "(a, c) \<in> r\<^sup>*"
OAlist_tc_lookup_def: fixes xs :: "('a, 'b) oalist_tc" shows "OAlist_tc_lookup xs = lookup_pair_tc (list_of_oalist_tc xs)"
SUP_least: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and u :: "'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq (f i) u" shows "less_eq (Sup (f ` A)) u"
has_derivative_transform: fixes x :: "'c"   and s :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'd" assumes "x \<in> s"   and "\<And>x. x \<in> s \<Longrightarrow> g x = f x"   and "(f has_derivative f') (at x within s)" shows "(g has_derivative f') (at x within s)"
sub_K_validity_good: fixes sub_K :: "msg"   and A :: "agent"   and B :: "agent"   and L :: "nat"   and K :: "nat"   and evs :: "event list" assumes "sub_K = Crypt (priEK A) \<lbrace>Number f_sub, Agent B, Nonce L, Key K\<rbrace>"   and "sub_K \<in> parts (knows Spy evs)"   and "A \<notin> bad"   and "evs \<in> zg" shows "Says A TTP \<lbrace>Number f_sub, Agent B, Nonce L, Key K, sub_K\<rbrace> \<in> set evs"
Says_imp_used: fixes A :: "agent"   and B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Says A B X \<in> set evs" shows "X \<in> used evs"
e_type_callcl: fixes \<S> :: "s_context"   and \<C> :: "t_context"   and cl :: "cl"   and t1s' :: "t list"   and t2s' :: "t list" assumes "\<S>\<bullet>\<C> \<turnstile> [Callcl cl] : t1s' _> t2s'" shows "\<exists>t1s t2s ts_c. t1s' = ts_c @ t1s \<and> t2s' = ts_c @ t2s \<and> cl_type cl = t1s _> t2s"
elliptic_isometryD_1: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "elliptic_isometry f" shows "bounded {(f ^^ n) x |n. True}"
nn_integral_empty: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" assumes "space M = {}" shows "integral\<^sup>N M f = 0"
ent_pure_pre_iff: fixes b :: "bool"   and Q :: "assn" shows "(emp * \<up> b \<Longrightarrow>\<^sub>A Q) = (b \<longrightarrow> (emp \<Longrightarrow>\<^sub>A Q))"
count_of_empty: fixes x :: "'b"   and xs :: "('b \<times> nat) list" assumes "x \<notin> fst ` set xs" shows "count_of xs x = 0"
enum_def: fixes G :: "('c, 'd) pre_digraph"   and s :: "'c"   and num :: "'c \<Rightarrow> nat"   and parent_edge :: "'c \<Rightarrow> 'd option"   and dist :: "'c \<Rightarrow> ereal"   and v :: "'c" assumes "shortest_paths_locale_step1 G s num parent_edge dist" shows "shortest_paths_locale_step1.enum num dist v = (if dist v = \<infinity> \<or> dist v = - \<infinity> then \<infinity> else enat (num v))"
absolutely_integrable_continuous: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "continuous_on (cbox a b) f" shows "f absolutely_integrable_on cbox a b"
greaterThanLessThan_subseteq_greaterThanLessThan: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.dense_linorder less_eq less" shows "(ord.greaterThanLessThan less a b \<subseteq> ord.greaterThanLessThan less c d) = (less a b \<longrightarrow> less_eq c a \<and> less_eq b d)"
t1_or_Hausdorff_space: fixes X :: "'c topology" shows "(t1_space X \<or> Hausdorff_space X) = t1_space X"
G_is_essentially_surjective: shows "essentially_surjective_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G"
simps_5: fixes uu :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and b :: "bexp"   and C :: "acom" shows "strip ({uu} WHILE b DO C) = WHILE b DO strip C"
pred_funI: fixes A :: "'a \<Rightarrow> bool"   and B :: "'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x. A x \<Longrightarrow> B (f x)" shows "pred_fun A B f"
maintain_S_precise_and_connected: fixes u :: "'v" assumes "u \<notin> S"   and "\<forall>v. v \<notin> S \<longrightarrow> D u \<le> D v" shows "D u = \<delta> s u"
foundation24: fixes \<tau> :: "'a state \<times> 'a state"   and X :: "'a state \<times> 'a state \<Rightarrow> 'b"   and Y :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "(\<tau> \<Turnstile> not (X \<triangleq> Y)) = (X \<tau> \<noteq> Y \<tau>)"