hcju/isabelleps huggingface dataset:
induct: fixes P :: "'b common_primitive match_expr \<Rightarrow> bool"   and a0 :: "'b common_primitive match_expr" assumes "P MatchAny"   and "\<And>uu_. P (Match (Dst_Ports (L4Ports uu_ [])))"   and "\<And>uv_ uw_. P (Match (Dst_Ports (L4Ports uv_ [uw_])))"   and "\<And>v vb va vd. P (Match (Dst_Ports (L4Ports v (vb # va # vd))))"   and "\<And>v. P (Match (Src v))"   and "\<And>v. P (Match (Dst v))"   and "\<And>v. P (Match (IIface v))"   and "\<And>v. P (Match (OIface v))"   and "\<And>v. P (Match (Prot v))"   and "\<And>v. P (Match (Src_Ports v))"   and "\<And>v. P (Match (MultiportPorts v))"   and "\<And>v. P (Match (L4_Flags v))"   and "\<And>v. P (Match (CT_State v))"   and "\<And>v. P (Match (Extra v))"   and "\<And>uz_. P (MatchNot (Match (Dst_Ports uz_)))"   and "\<And>v. P (MatchNot (Match (Src v)))"   and "\<And>v. P (MatchNot (Match (Dst v)))"   and "\<And>v. P (MatchNot (Match (IIface v)))"   and "\<And>v. P (MatchNot (Match (OIface v)))"   and "\<And>v. P (MatchNot (Match (Prot v)))"   and "\<And>v. P (MatchNot (Match (Src_Ports v)))"   and "\<And>v. P (MatchNot (Match (MultiportPorts v)))"   and "\<And>v. P (MatchNot (Match (L4_Flags v)))"   and "\<And>v. P (MatchNot (Match (CT_State v)))"   and "\<And>v. P (MatchNot (Match (Extra v)))"   and "\<And>m1 m2. \<lbrakk>P m1; P m2\<rbrakk> \<Longrightarrow> P (MatchAnd m1 m2)"   and "\<And>vb_ vc_. P (MatchNot (MatchAnd vb_ vc_))"   and "\<And>vd_. P (MatchNot (MatchNot vd_))"   and "P (MatchNot MatchAny)" shows "P a0"
norm_one: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool" assumes "class.real_normed_algebra_1 plus zero minus uminus scaleR times one dist norm sgn uniformity open" shows "norm one = 1"
pcancel_into_cancel: fixes xs :: "'d list"   and ys :: "'d list" assumes "pcancel1 xs ys" shows "cancel xs ys"
split_aforms_largest_uncond_def: fixes X :: "('a \<times> 'a pdevs) list" shows "split_aforms_largest_uncond X = (let (i, x) = max_pdev (abssum_of_pdevs_list (map snd X)) in split_aforms X i)"
incseq_tendsto_limsup: fixes f :: "nat \<Rightarrow> 'a" assumes "incseq f" shows "f \<longlonglongrightarrow> limsup f"
map_butlast: fixes f :: "'f \<Rightarrow> 'e"   and xs :: "'f list" shows "map f (butlast xs) = butlast (map f xs)"
finite_dimension_imp_algebraic: fixes K :: "'a set"   and F :: "'a set"   and x :: "'a" assumes "subfield K R"   and "subring F R"   and "finite_dimension K F"   and "x \<in> F" shows "(algebraic over K) x"
gR_rel_mult: fixes x :: "'c"   and y :: "'c" assumes "D x y" shows "x \<preceq>\<^sub>R x \<oplus> y"
union_eq_empty: fixes M :: "'b multiset"   and N :: "'b multiset" shows "(M + N = \<Empt>) = (M = \<Empt> \<and> N = \<Empt>)"
the_paramR: fixes r :: "'b option"   and l :: "'c option"   and R :: "('c \<times> 'b) set" assumes "r \<noteq> None"   and "(l, r) \<in> \<langle>R\<rangle>option_rel" shows "(the l, the r) \<in> R"
butlast_take: fixes n :: "nat"   and xs :: "'d list" assumes "n \<le> length xs" shows "butlast (take n xs) = take (n - 1) xs"
choose_iapath: fixes u :: "'a"   and v :: "'a" assumes "\<exists>p. gen_iapath (verts3 (with_proj G)) u p v" shows "gen_iapath (verts3 (with_proj G)) u (choose_iapath u v) v"
eval_wlp_Skip: fixes P :: "'b \<Rightarrow> real" shows "wlp Skip P = P"
prems_1: shows "check_no_overlap2 \<gamma> (a # t)"
A: fixes p :: "nat" assumes "1 \<le> p" shows "Busemann_function_at xi ((f ^^ r p) basepoint) basepoint / real (r p) - (2 * deltaG TYPE('a) + 1) * (1 / real (r p)) \<le> 0"
greater_imp_iprev_ge: fixes b :: "nat"   and a :: "nat"   and I :: "nat set" assumes "b < a"   and "a \<in> I"   and "b \<in> I" shows "b \<le> iprev a I"
NSCLIM_NSCRLIM_iff: fixes f :: "'a \<Rightarrow> complex"   and x :: "'a"   and L :: "complex" shows "f \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S L = (\<lambda>y. cmod (f y - L)) \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S 0"
Norm_eq_lnorm: fixes f :: "'a \<Rightarrow> real"   and p :: "real"   and M :: "'a measure" assumes "f \<in> lspace M (ennreal p)"   and "0 < p" shows "Norm (\<LL> (ennreal p) M) f = lnorm M p f"
S_def: shows "S = {\<chi>. w' \<Turnstile> \<chi>}"
is_subset_1: fixes c :: "'a" assumes "c \<in> ta.Qi" shows "c \<in> ta.Q"
cases: fixes x :: "tm \<times> tm" obtains z :: "name" and t :: "tm" and u :: "tm" where "atom z \<sharp> (t, u)" and "x = (t, u)"
tm_lower_order_of_normed_range: fixes x :: "real"   and e :: "nat \<Rightarrow> real"   and t :: "taylor_model"   and a :: "float interval list"   and I :: "float interval list"   and prec :: "nat"   and ord :: "nat" assumes "x \<in>\<^sub>i range_tm e t"   and "develops_at_within e a I"   and "num_params (tm_poly t) \<le> length I" shows "x \<in>\<^sub>i range_tm e (tm_lower_order_of_normed prec ord I a t)"
sorted_append: fixes xs :: "'d Word.word list"   and ys :: "'d Word.word list" shows "signed.sorted (xs @ ys) = (signed.sorted xs \<and> signed.sorted ys \<and> (\<forall>x\<in>set xs. \<forall>y\<in>set ys. x \<le>s y))"
redT_updTs_new_thread_ts: fixes ts :: "'a \<Rightarrow> ('b \<times> 'c \<Rightarrow>f nat) option"   and tas :: "('a, 'b, 'd) new_thread_action list"   and t :: "'a"   and x :: "'b"   and m :: "'d" assumes "thread_oks ts tas"   and "NewThread t x m \<in> set tas" shows "redT_updTs ts tas t = \<lfloor>(x, no_wait_locks)\<rfloor>"
conga_pseudo_refl: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "A \<noteq> B"   and "C \<noteq> B" shows "A B C CongA C B A"
assms_5: fixes U :: "'x set"   and S :: "('x \<Rightarrow>\<^sub>0 nat) set"   and t :: "'x \<Rightarrow>\<^sub>0 nat" assumes "U \<subseteq> X"   and "finite S"   and "0 \<notin> S"   and "S \<inter> .[U] = {}" shows "P t U S ([], [(monomial (1::'a) t, U)])"
fold_congs_2: fixes r :: "('c, 'd, 'e) automaton_scheme"   and r' :: "('c, 'd, 'e) automaton_scheme"   and v' :: "('c \<times> 'd \<times> 'c) set"   and f :: "('c \<times> 'd \<times> 'c) set \<Rightarrow> ('c \<times> 'd \<times> 'c) set"   and f' :: "('c \<times> 'd \<times> 'c) set \<Rightarrow> ('c \<times> 'd \<times> 'c) set" assumes "r = r'"   and "automaton.trans r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "trans_update f r = trans_update f' r'"
iASSUME_def: fixes return :: "unit \<Rightarrow> 'a"   and \<Phi> :: "bool" shows "iASSUME return \<Phi> \<equiv> if \<Phi> then return () else bot"
simps_3: fixes x :: "real"   and y :: "real" shows "real_min_top.R x + real_min_top.R y = real_min_top.R (min x y)"
uq: shows "Mp (u * q) = u * q - smult m (Dp (u * q))"
subtensor_tensor_prod: fixes A :: "'a tensor"   and i :: "nat"   and B :: "'a tensor" assumes "dims A \<noteq> []"   and "i < hd (dims A)" shows "subtensor (local.tensor_prod_otimes A B) i = local.tensor_prod_otimes (subtensor A i) B"
simps: fixes \<Sigma> :: "'a set set"   and \<pi> :: "'a ltl \<Rightarrow> nat option"   and \<chi> :: "'a ltl" shows "Acc_fin \<Sigma> \<pi> \<chi> = \<Union> (embed_transition_snd ` \<Union> (\<upharpoonleft>\<^sub>\<chi> ` (mojmir_to_rabin_def.fail\<^sub>R \<Sigma> \<delta>\<^sub>M (q\<^sub>0\<^sub>M (theG \<chi>)) (Collect (ltl_prop_entails_abs (dom \<pi>))) \<union> mojmir_to_rabin_def.merge\<^sub>R \<delta>\<^sub>M (q\<^sub>0\<^sub>M (theG \<chi>)) (Collect (ltl_prop_entails_abs (dom \<pi>))) (the (\<pi> \<chi>)))))"
2: shows "comparator_of (lookup_pp (rep_nat_pp (PP_oalist xs)) x) (lookup_pp (rep_nat_pp (PP_oalist ys)) x) = lex_comp' (PP_oalist xs) (PP_oalist ys)"
product_cone_def: fixes J :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'b \<Rightarrow> 'c"   and a :: "'c"   and \<pi> :: "'b \<Rightarrow> 'c" shows "product_cone J C D a \<pi> \<equiv> (category J \<and> category C) \<and> discrete_diagram J C D \<and> limit_cone J C D a \<pi>"
sup0: fixes a :: "'c" shows "a \<equiv> a \<squnion> bot"
extreme_point_not_in_REL_INTERIOR: fixes x :: "'b"   and S :: "'b set" assumes "x extreme_point_of S"   and "S \<noteq> {x}" shows "x \<notin> rel_interior S"
subset_imp_ensures: fixes A :: "'d set"   and B :: "'d set"   and F :: "'d program" assumes "A \<subseteq> B" shows "F \<in> A ensures B"
monad_fail_alt_envT: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and fail :: "'b"   and alt :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "monad_fail_alt return bind fail alt" shows "monad_fail_alt (return_env return) (bind_env bind) (fail_env fail) (alt_env alt)"
iso_unit_2: shows "Left_a.in_hom (some_unit a) (a \<star> a) a"
ev_ex_nxt: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream" assumes "ev \<phi> xs" shows "\<exists>n. (nxt ^^ n) \<phi> xs"
aha: shows "card (fst ` {(x, y) |x. x < y in xs \<and> y < x in ys}) = card {(x, y) |x. x < y in xs \<and> y < x in ys}"
affine_extension2_def: fixes fnctn_aff :: "nat \<Rightarrow> 'd \<times> 'd pdevs \<Rightarrow> 'e \<times> 'e pdevs \<Rightarrow> ('f \<times> 'f pdevs) option"   and fnctn :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" shows "affine_extension2 fnctn_aff fnctn = (\<forall>d a1 a2 X e2. fnctn_aff d a1 a2 = Some X \<longrightarrow> e2 \<in> UNIV \<rightarrow> {- 1..1} \<longrightarrow> degree_aform a1 \<le> d \<longrightarrow> degree_aform a2 \<le> d \<longrightarrow> (\<exists>e3\<in>UNIV \<rightarrow> {- 1..1}. fnctn (aform_val e2 a1) (aform_val e2 a2) = aform_val e3 X \<and> (\<forall>n<d. e3 n = e2 n) \<and> aform_val e2 a1 = aform_val e3 a1 \<and> aform_val e2 a2 = aform_val e3 a2))"
get_shadow_root_ptr_in_heap: fixes h :: "(_) heap"   and element_ptr :: "(_) element_ptr"   and children :: "(_) shadow_root_ptr option" assumes "h \<turnstile> get_shadow_root element_ptr \<rightarrow>\<^sub>r children" shows "element_ptr |\<in>| element_ptr_kinds h"
dsn_rrep': fixes i :: "nat" shows "paodv i \<TTurnstile> (recvmsg (\<lambda>m. rreq_rrep_sn m \<and> msg_zhops m) \<rightarrow>) onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l \<in> {PAodv-:6, PAodv-:7} \<union> {PRrep-:n |n. True} \<longrightarrow> 1 \<le> dsn \<xi>)"
pmf_positive_iff: fixes p :: "'b pmf"   and x :: "'b" shows "(0 < pmf p x) = (x \<in> set_pmf p)"
meet_aboveI: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z \<sqsubseteq> x"   and "z \<sqsubseteq> y" shows "z \<sqsubseteq> x \<sqinter> y"
Cons_snoc: fixes x :: "'a"   and ys :: "'a list"   and y :: "'a" assumes "xs = x # ys @ [y]" shows "P"
inv_inv: fixes G :: "('e, 'f) monoid_scheme"   and x :: "'e" assumes "Group.group G"   and "x \<in> carrier G" shows "inv\<^bsub>G\<^esub> (inv\<^bsub>G\<^esub> x) = x"
monom_rep_X_pow: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and n :: "nat" assumes "UP_cring R"   and "a \<in> carrier R" shows "monom (UP R) a n = a \<odot>\<^bsub>UP R\<^esub> X_poly R [^]\<^bsub>UP R\<^esub> n"
is_canon_rewriter_def: fixes rword :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and A :: "('t \<Rightarrow>\<^sub>0 'b) set"   and u :: "'t"   and p :: "'t \<Rightarrow>\<^sub>0 'b" shows "is_canon_rewriter rword A u p = (p \<in> A \<and> p \<noteq> 0 \<and> lt p adds\<^sub>t u \<and> (\<forall>a\<in>A. a \<noteq> 0 \<longrightarrow> lt a adds\<^sub>t u \<longrightarrow> rword (spp_of a) (spp_of p)))"
jm: shows "k \<cdot>\<^sub>m jordan_matrix (na # n_as) = four_block_mat (k \<cdot>\<^sub>m jordan_block n a) (0\<^sub>m n (sum_list (map fst n_as))) (0\<^sub>m (sum_list (map fst n_as)) n) (k \<cdot>\<^sub>m jordan_matrix n_as)"
ssubst: fixes t :: "'g"   and s :: "'g"   and P :: "'g \<Rightarrow> bool" assumes "t = s"   and "P s" shows "P t"
rank_0I: fixes n :: "nat"   and nc :: "nat" shows "vec_space.rank n (0\<^sub>m n nc) = 0"
simps: fixes c :: "(nat, 'a \<times> QDelta) mapping \<times> (nat, 'a \<times> QDelta) mapping"   and cs :: "QDelta ns_constraint list"   and asi_tv :: "('a, ('a \<times> QDelta atom) list) mapping \<times> ((nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping) \<times> 'a list"   and s :: "('a, QDelta) state" shows "backtrack_simplex (Simplex_Checkpoint c) (Simplex_State (cs, asi_tv, s)) = Simplex_State (cs, asi_tv, backtrack_s c s)"
trelRS: shows "rel_reflects_barb_set TRel TWB {success}"
distinct_25: fixes x31 :: "nat set"   and x32 :: "nat set"   and x33 :: "'d"   and x51 :: "nat"   and x52 :: "nat" shows "x31\<not>x32:arrive(x33) \<noteq> disconnect(x51, x52)"
Erdos_Milner: fixes \<nu> :: "V"   and n :: "nat" assumes "\<nu> \<in> elts \<omega>1" shows "partn_lst_VWF (\<omega> \<up> (1 + \<nu> * ord_of_nat n)) [ord_of_nat (2 ^ n), \<omega> \<up> (1 + \<nu>)] 2"
bernoulli_denom_1: shows "bernoulli_denom 1 = 2"
only_new_introduces_lids': fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and r :: "'a"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and s' :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and l :: "'b" assumes "substitution subst"   and "substitution.revision_step subst r s s'"   and "\<And>\<sigma> \<tau> \<E> v. s r \<noteq> Some (\<sigma>, \<tau>, \<E> [Ref (VE v)])"   and "l \<notin> LID\<^sub>G s" shows "l \<notin> LID\<^sub>G s'"
simps_10: fixes e :: "expr"   and Copt :: "char list option"   and M :: "char list"   and es :: "expr list" shows "fv (Call e Copt M es) = fv e \<union> fvs es"
wf\<^sub>M\<^sub>G\<^sub>U_def: fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and s :: "('b, 'a) Term.term"   and t :: "('b, 'a) Term.term" shows "wf\<^sub>M\<^sub>G\<^sub>U \<theta> s t \<equiv> wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta> \<and> MGU \<theta> s t \<and> subst_domain \<theta> \<union> range_vars \<theta> \<subseteq> fv s \<union> fv t"
Rstep_TrueR: fixes j :: "nat"   and A :: "('sf, 'sc, 'sz) formula list"   and S :: "('sf, 'sc, 'sz) formula list" shows "Rrule_result TrueR j (A, S) = []"
that: fixes r :: "V"   and A :: "V" assumes "x = \<langle>\<langle>r, A\<rangle>, r -`\<^sub>\<circ> A\<rangle>"   and "r \<in>\<^sub>\<circ> R"   and "A \<in>\<^sub>\<circ> X" shows "thesis"
simps_2: fixes xs :: "'h list"   and opt :: "('f, 'g, 'h) env option"   and b :: "'g"   and es :: "'h \<Rightarrow> ('f, 'g, 'h) env option" shows "update xs opt (Env b es) = (case xs of [] \<Rightarrow> case opt of None \<Rightarrow> Env b es | Some e \<Rightarrow> e | y # ys \<Rightarrow> Env b (es(y := ??.Nested_Environment.update_option ys opt (es y))))"
pol_polC: fixes \<sigma> :: "'tp"   and P :: "'psym" shows "pol \<sigma> P = polC P"
functor_axioms: shows "functor (\<cdot>\<^sub>B) (\<cdot>\<^sub>C) F"
simps_1: fixes x :: "'b"   and y :: "'b" shows "srcNet (DenyAllFromTo x y) = x"
i_set_indp_i_set_ind_eq: shows "i_set_indp = (\<lambda>x. x \<in> i_set_ind)"
prems_1: shows "\<forall>i<Suc k. eval (gs ! i) xs \<down>"
frontier_cball: fixes a :: "'b"   and e :: "real" shows "frontier (cball a e) = sphere a e"
steps_Stuck_prop: fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and c :: "('a, 'b, 'c) com"   and s :: "('a, 'c) xstate"   and c' :: "('a, 'b, 'c) com"   and s' :: "('a, 'c) xstate" assumes "\<Gamma>\<turnstile> (c, s) \<rightarrow>\<^sup>* (c', s')"   and "s = Stuck" shows "s' = Stuck"
bounds_consistent_lt_lb: fixes s :: "('a, 'b) state"   and c :: "'b"   and x :: "nat" assumes "\<diamond> s"   and "c >\<^sub>u\<^sub>b \<B>\<^sub>u s x" shows "\<not> c <\<^sub>l\<^sub>b \<B>\<^sub>l s x"
hyps: shows "set_pmf (bv n) = {x. length x = n}"
distinct_1: fixes x1 :: "nat"   and x2 :: "nat" shows "Regions.intv.Const x1 \<noteq> Regions.intv.Intv x2"
min_def: fixes a :: "'a"   and b :: "'a" shows "min a b = (if a \<le> b then a else b)"
linorder_less_linear: fixes x :: "'a"   and y :: "'a" shows "x < y \<or> x = y \<or> y < x"
list_seq_indexE_length: fixes seq :: "nat \<Rightarrow> 'c list"   and i :: "nat" assumes "\<And>i. \<exists>x. seq (Suc i) = x # seq i" obtains j :: "nat" where "i < length (seq j)"
x_2: shows "infinite {n. birkhoff_sum f n x \<in> {0..1}}"
inverse_arrowsE: fixes f :: "'c"   and g :: "'c" assumes "C.inverse_arrows f g" obtains "C.ide (g \<cdot>\<^sub>C f)" and "C.ide (f \<cdot>\<^sub>C g)"
hull_closed_lists: fixes us :: "'b list list"   and G :: "'b list set" assumes "us \<in> lists \<langle>G\<rangle>" shows "concat us \<in> \<langle>G\<rangle>"
simps_71: fixes x4 :: "val"   and x31 :: "char list"   and x32 :: "expr" shows "Val x4 \<noteq> \<lparr>x31\<rparr>x32"
l_set_mode_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_mode :: "(_) shadow_root_ptr \<Rightarrow> shadow_root_mode \<Rightarrow> ((_) heap, exception, unit) prog"   and set_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_set_mode type_wf set_mode set_mode_locs \<equiv> ((\<forall>shadow_root_ptr shadow_root_mode h h'. writes (set_mode_locs shadow_root_ptr) (set_mode shadow_root_ptr shadow_root_mode) h h') \<and> (\<forall>h shadow_root_ptr shadow_root_mode. type_wf h \<longrightarrow> shadow_root_ptr |\<in>| shadow_root_ptr_kinds h \<longrightarrow> h \<turnstile> ok set_mode shadow_root_ptr shadow_root_mode)) \<and> (\<forall>h shadow_root_ptr shadow_root_mode. h \<turnstile> ok set_mode shadow_root_ptr shadow_root_mode \<longrightarrow> shadow_root_ptr |\<in>| shadow_root_ptr_kinds h) \<and> (\<forall>w shadow_root_ptr h h'. w \<in> set_mode_locs shadow_root_ptr \<longrightarrow> h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> object_ptr_kinds h = object_ptr_kinds h') \<and> (\<forall>w shadow_root_ptr h h'. w \<in> set_mode_locs shadow_root_ptr \<longrightarrow> h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> type_wf h = type_wf h')"
continuous_on_generate_topology: fixes X :: "'b set set"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b" assumes "open = generate_topology X"   and "\<And>B. B \<in> X \<Longrightarrow> \<exists>C. open C \<and> C \<inter> A = f -` B \<inter> A" shows "continuous_on A f"
IH_1: assumes "Broadcast m1 \<in> set (history i)"   and "Broadcast m2 \<in> set (history i)" shows "Broadcast m1 \<sqsubset>\<^sup>i Broadcast m2"
rotation_exists: fixes a :: "(real, 'a) vec"   and b :: "(real, 'a) vec" assumes "2 \<le> CARD('a)"   and "norm a = norm b" obtains f :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec" where "orthogonal_transformation f" and "det (matrix f) = 1" and "f a = b"
cont_is_mono: fixes f :: "'g \<Rightarrow> 'h" assumes "cont f" shows "mono f"
suc_inField: fixes B :: "'a set" assumes "B \<subseteq> Field r"   and "local.AboveS B \<noteq> {}" shows "suc B \<in> Field r"
exec_step_ind_cases_14: fixes P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap"   and \<sigma> :: "jvm_state" assumes "exec_step_ind (StepI IAdd) P h stk loc C M pc ics frs sh \<sigma>" obtains "\<sigma> = (None, h, (Intg (the_Intg (hd (tl stk)) + the_Intg (hd stk)) # tl (tl stk), loc, C, M, Suc pc, ics) # frs, sh)"
QBZ: shows "(Q + B) \<inter> \<int>\<^sub>v \<subseteq> carrier_vec n"
lt_alt: fixes p :: "'a \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0" shows "punit.lt p = ordered_powerprod_lin.Max (keys p)"
l_null: fixes x :: "'a list set" assumes "x \<in> carrier (Rupt K p)" shows "\<zero>\<^bsub>Rupt K p\<^esub> \<otimes>\<^bsub>Rupt K p\<^esub> x = \<zero>\<^bsub>Rupt K p\<^esub>"
verit_sko_ex': fixes P :: "'a \<Rightarrow> bool"   and A :: "bool" assumes "P (SOME x. P x) = A" shows "(\<exists>x. P x) = A"
simps_217: fixes x141 :: "expr"   and x142 :: "expr"   and x121 :: "expr"   and x122 :: "expr" shows "while (x141) x142 \<noteq> x121;; x122"
simps_7: fixes cl :: "('test, 'atom, 'choice) cmd list" shows "brn (ParT cl) = lsum brn cl"
q_delta: shows "ego.q \<delta> = ego2.s 0"
mbind'_bind: fixes a :: "'g"   and S :: "'g list"   and F :: "'g \<Rightarrow> 'd \<Rightarrow> ('f \<times> 'd) option"   and M :: "'f list \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'd) option" shows "bind\<^sub>S\<^sub>E (mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>S\<^sub>t\<^sub>o\<^sub>p (a # S) F) M = _bind_SE a (F a) (_bind_SE x (mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>S\<^sub>t\<^sub>o\<^sub>p S F) (M (a # x)))"
f: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list"   and t :: "state"   and m :: "'c match_expr"   and c :: "char list" assumes "\<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>r t"   and "matches \<gamma> m p"   and "all_chains (no_call_to c) \<Gamma> rs" shows "\<Gamma>(c \<mapsto> rs),\<gamma>,p\<turnstile> [Rule m (Call c)] \<Rightarrow>\<^sub>z t"
AssertLiteralStartQIreleveant: fixes state :: "State"   and literal :: "Literal"   and decision :: "bool"   and Q' :: "Literal list"   and Q'' :: "Literal list" assumes "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)" shows "let state' = assertLiteral literal decision (state\<lparr>getQ := Q'\<rparr>); state'' = assertLiteral literal decision (state\<lparr>getQ := Q''\<rparr>) in getM state' = getM state'' \<and> getF state' = getF state'' \<and> getSATFlag state' = getSATFlag state'' \<and> getConflictFlag state' = getConflictFlag state''"
keys_mult: fixes f :: "'a \<Rightarrow>\<^sub>0 'b"   and g :: "'a \<Rightarrow>\<^sub>0 'b" shows "keys (f * g) \<subseteq> {a + b |a b. a \<in> keys f \<and> b \<in> keys g}"
simps_14: fixes ipassmt :: "iface \<Rightarrow> ('a word \<times> nat) list option"   and m1 :: "'a common_primitive match_expr"   and m2 :: "'a common_primitive match_expr" shows "iiface_rewrite ipassmt (MatchAnd m1 m2) = MatchAnd (iiface_rewrite ipassmt m1) (iiface_rewrite ipassmt m2)"
admits_SNF: shows "\<forall>A n. A \<in> carrier_mat n n \<and> isDiagonal_mat A \<longrightarrow> (\<exists>P Q. P \<in> carrier_mat n n \<and> Q \<in> carrier_mat n n \<and> invertible_mat P \<and> invertible_mat Q \<and> Smith_normal_form_mat (P * A * Q))"
dis_def: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<psi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list" shows "dis \<phi> \<psi> \<pi>l \<equiv> \<phi> \<pi>l \<or> \<psi> \<pi>l"
assms_2: fixes x :: "complex" assumes "x \<in> S" shows "(f has_field_derivative f' x) (at x within S)"
intro: fixes relation :: "('b \<times> 'b) set"   and carrier :: "'b set" assumes "\<And>x y. x \<succeq>[relation] y \<Longrightarrow> x \<in> carrier"   and "\<And>x y. x \<succeq>[relation] y \<Longrightarrow> y \<in> carrier"   and "preorder_on carrier relation" shows "preference carrier relation"
mult_bounds_enclose_zero1_2: fixes la :: "'b"   and ua :: "'b"   and lb :: "'b"   and ub :: "'b" assumes "la \<le> (0::'b)"   and "(0::'b) \<le> ua" shows "(0::'b) \<le> max (la * lb) (max (la * ub) (max (lb * ua) (ua * ub)))"
fresh_PairD_2: fixes a :: "atom"   and x :: "'a"   and y :: "'b" assumes "a \<sharp> (x, y)" shows "a \<sharp> y"
msb_word_iff_bit: fixes w :: "'a word" shows "msb w = bit w (LENGTH('a) - Suc 0)"
assms_1: fixes i :: "nat" shows "algebraic_int (coeff p i)"
surj_on_minus: fixes y :: "nat"   and q :: "nat" assumes "inj_on (\<lambda>b. (y + (q - b)) mod q) {..<q}" shows "(\<lambda>b. (y + (q - b)) mod q) ` {..<q} = {..<q}"
history_inv_def: fixes ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m :: "nat \<Rightarrow> nat" shows "history_inv ts m \<equiv> valid_history program_step ts \<and> valid_program_history ts \<and> valid_reads m ts"