hcju/isabelleps huggingface dataset:
step_preserves_length_ts: fixes memop_step :: "instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b \<times> (nat \<Rightarrow> nat) \<times> 'c \<times> 'd \<times> 'e \<times> 'f \<Rightarrow> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b \<times> (nat \<Rightarrow> nat) \<times> 'c \<times> 'd \<times> 'e \<times> 'f \<Rightarrow> bool"   and storebuffer_step :: "(nat \<Rightarrow> nat) \<times> 'b \<times> 'd \<times> 'e \<times> 'f \<Rightarrow> (nat \<Rightarrow> nat) \<times> 'b \<times> 'd \<times> 'e \<times> 'f \<Rightarrow> bool"   and program_step :: "(nat \<Rightarrow> nat option) \<Rightarrow> 'g \<Rightarrow> 'g \<times> instr list \<Rightarrow> bool"   and "record" :: "'g \<Rightarrow> 'g \<Rightarrow> instr list \<Rightarrow> 'b \<Rightarrow> 'b"   and ts :: "('g \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b \<times> 'c \<times> 'd \<times> 'e) list"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "'f"   and ts' :: "('g \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b \<times> 'c \<times> 'd \<times> 'e) list"   and m' :: "nat \<Rightarrow> nat"   and \<S>' :: "'f" assumes "computation.concurrent_step memop_step storebuffer_step program_step record (ts, m, \<S>) (ts', m', \<S>')" shows "length ts' = length ts"
injectivity: fixes x :: "'a mod_ring poly"   and y :: "'a mod_ring poly" assumes "to_int_poly x = to_int_poly y" shows "x = y"
eq1: shows "lookup_dflt (map_pair f xs) (fst (f (k, v))) = snd (f (k, lookup_dflt xs k))"
thetaSAtm_Sbis: fixes atm :: "'atom" assumes "compatAtm atm" shows "thetaSAtm atm \<subseteq> Sbis"
assms: assumes "\<Phi>" shows "ASSUME \<Phi> \<bind> (\<lambda>_. S) \<le> \<Down> R S'"
emb_eq: shows "PF.emb UNIV J (Pi\<^sub>E J F') = PF.emb UNIV {0..<Suc n} (Pi\<^sub>E {0..<Suc n} F)"
drop_i_f_join_eq2: fixes I :: "nat set"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "finite I"   and "card I \<le> n" shows "f \<Join>\<^sub> I \<up> n = []"
derived_S5_rules_1_b: fixes \<phi> :: "\<o>"   and \<psi> :: "\<o>"   and v :: "i" assumes "\<And>v. [\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<psi> in v]" shows "[\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<psi> in v]"
map_ugraph_wellformed: fixes G :: "nat set \<times> nat set set"   and f :: "nat \<Rightarrow> nat" assumes "uwellformed G"   and "inj_on f (uverts G)" shows "uwellformed (map_ugraph f G)"
that: fixes us :: "'a list"   and vs :: "'a list" assumes "xs = us @ x # vs"   and "x \<notin> set us"   and "remdups_fwd_acc Acc us = ys"   and "remdups_fwd_acc (Acc \<union> set ys \<union> {x}) vs = zs" shows "thesis"
emeasure_lborel_Icc: fixes l :: "real"   and u :: "real" assumes "l \<le> u" shows "emeasure lborel {l..u} = ennreal (u - l)"
comparator_sum_simps_4: fixes comp\<^sub>'\<^sub>a :: "'c \<Rightarrow> 'c \<Rightarrow> order"   and comp\<^sub>'\<^sub>b :: "'d \<Rightarrow> 'd \<Rightarrow> order"   and x :: "'d"   and ya :: "'d" shows "comparator_sum comp\<^sub>'\<^sub>a comp\<^sub>'\<^sub>b (Inr x) (Inr ya) = comp\<^sub>'\<^sub>b x ya"
assms: shows "low_equal s1 s2 \<and> t1 = dcache_mod c v s1 \<and> t2 = dcache_mod c v s2"
functor_axioms: shows "Functor F : AA \<longrightarrow> BB"
semilattice_sup_ow_axioms: shows "semilattice_sup_ow U\<^sub>1 sup\<^sub>1 (\<le>\<^sub>o\<^sub>w\<^sub>.\<^sub>1) (<\<^sub>o\<^sub>w\<^sub>.\<^sub>1)"
Stuck_end: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"   and "s = Stuck" shows "t = Stuck"
idx_pm_of_pm_def: fixes xs :: "'d list"   and f :: "'d \<Rightarrow>\<^sub>0 'c" shows "idx_pm_of_pm xs f = Abs_poly_mapping (\<lambda>i. lookup f (xs ! i) when i < length xs)"
simps_13: fixes bs :: "real list"   and p :: "fm"   and q :: "fm" shows "Ifm bs (Iff p q) = (Ifm bs p = Ifm bs q)"
set_less_aux_antisym: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and B :: "'a set" assumes "ord.set_less_aux less_eq A B"   and "ord.set_less_aux less_eq B A" shows "False"
assms: shows "substitutes (SimplyTyped.Pair A B) x M X"
simps_2: fixes f :: "'c"   and Tl :: "'c trm list" shows "gT (Fn f Tl) = Fn (Oldf f) (map gT Tl)"
trans: fixes b :: "'g word"   and a :: "'g word"   and c :: "'g word" assumes "b \<le>s a"   and "c \<le>s b" shows "c \<le>s a"
repelling_fixed_point_def: fixes f :: "'b \<Rightarrow> 'b" shows "repelling_fixed_point f = (SOME xi. xi \<in> Gromov_boundary \<and> Gromov_extension f xi = xi \<and> 0 < additive_strength f xi)"
ide_compE: fixes g :: "'a"   and f :: "'a" assumes "ide (g \<cdot> f)" obtains "seq g f" and "seq f g" and "g \<cdot> f = local.dom f" and "g \<cdot> f = cod g"
filter: fixes p' :: "'a multiset" assumes "p' \<in># remove1_mset p (mset_set (mset_set ` P))" shows "p \<noteq> p'"
sup2E: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d"   and y :: "'e" assumes "sup A B x y" obtains "A x y"   | "B x y"
adds_refl: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a" assumes "class.comm_monoid_add plus zero" shows "plus.adds plus a a"
count_greater_zero_iff: fixes M :: "'c multiset"   and x :: "'c" shows "(0 < count M x) = (x \<in># M)"
dghm_length: shows "vcard \<GG> = 4\<^sub>\<nat>"
code: fixes xs :: "'b llist"   and ys :: "'b llist" shows "xs @@ ys = (case xs of LNil \<Rightarrow> ys | x ## xs' \<Rightarrow> x ## xs' @@ ys)"
Pi_split_domain: fixes x :: "'b \<Rightarrow> 'c"   and I :: "'b set"   and J :: "'b set"   and X :: "'b \<Rightarrow> 'c set" shows "(x \<in> Pi (I \<union> J) X) = (x \<in> Pi I X \<and> x \<in> Pi J X)"
flip: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "porder le" shows "porder (\<lambda>x y. le y x)"
subst_range_subst_Fun_single: fixes x :: "'b"   and f :: "'a"   and T :: "('a, 'b) Term.term list" shows "subst_range (Var(x := Fun f T)) = {Fun f T}"
B_def: shows "B = (\<lambda>X. Abs_nonzero ` (X - {0::'a})) ` A"
finite_dimensionI: fixes n :: "nat"   and K :: "'a set"   and E :: "'a set" assumes "dimension n K E" shows "finite_dimension K E"
dvd1: shows "k dvd t1 * int u2 - t2 * int u1"
wf_mbufn_take: fixes f :: "event_data option list set list \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and buf :: "event_data option list set list list"   and z' :: "'a"   and buf' :: "event_data option list set list list"   and i :: "nat"   and js :: "nat list"   and Ps :: "(nat \<Rightarrow> event_data option list set \<Rightarrow> bool) list" assumes "mbufn_take f z buf = (z', buf')"   and "wf_mbufn i js Ps buf" shows "wf_mbufn (Mini i js) js Ps buf'"
HSem_reorder_head: fixes x :: "'a"   and y :: "'a"   and ESem :: "'c \<Rightarrow> ('a \<Rightarrow> 'b) \<rightarrow> 'b"   and e1 :: "'c"   and e2 :: "'c"   and \<Gamma> :: "('a \<times> 'c) list"   and \<rho> :: "'a \<Rightarrow> 'b" assumes "x \<noteq> y" shows "has_ESem.HSem ESem ((x, e1) # (y, e2) # \<Gamma>)\<cdot>\<rho> = has_ESem.HSem ESem ((y, e2) # (x, e1) # \<Gamma>)\<cdot>\<rho>"
dim_eliminate_entries_gen_1: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and v :: "nat \<Rightarrow> 'c"   and B :: "'c mat"   and i :: "nat"   and as :: "nat" shows "dim_row (eliminate_entries_gen minus times v B i as) = dim_row B"
revis_extensionality: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a"   and \<psi> :: "'a" assumes "K = Cn A"   and "Cn {\<phi>} = Cn {\<psi>}" shows "K \<^bold>* \<phi> = K \<^bold>* \<psi>"
pairs_top_closed: shows "stone_phi.pairs_top \<in> stone_phi.pairs"
preserves_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>B b\<guillemotright>" shows "\<guillemotleft>trg\<^sub>B f : trg\<^sub>B a \<Rightarrow>\<^sub>B trg\<^sub>B b\<guillemotright>"
minus_divide_eq_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and 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 b :: "'b"   and c :: "'b"   and a :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "(uminus (divide b c) = a) = (if c \<noteq> zero then uminus b = times a c else a = zero)"
init_time: fixes f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "real set"   and S :: "'b set"   and t\<^sub>0 :: "real" assumes "picard_lindeloef f T S t\<^sub>0" shows "t\<^sub>0 \<in> T"
cdf_nondecreasing: fixes x :: "real"   and y :: "real" assumes "x \<le> y" shows "cdf M x \<le> cdf M y"
simps_5: fixes r :: "mregex"   and s :: "mregex" shows "RPD (MTimes r s) = MTimesL r (RPD s) \<union> RPD r"
set_def: fixes xs :: "['d]" shows "set xs = {x. listmem x xs}"
pseudo_proj_Suc_prefix: fixes n :: "nat"   and w :: "bool stream"   and x :: "bool stream" assumes "pseudo_proj_True n w = pseudo_proj_True n x" shows "pseudo_proj_True (Suc n) w \<in> {pseudo_proj_True n x, pseudo_proj_False n x}"
inner_zero_left: fixes x :: "'a" shows "inner_field (0::'a) x = (0::'a)"
closure_ball: fixes e :: "real"   and x :: "'b" assumes "0 < e" shows "closure (ball x e) = cball x e"
dom_eqI': fixes a :: "'b"   and f :: "'b" assumes "a \<in> B.domains f" shows "a = B.dom f"
VV_impl1: fixes v :: "'a"   and p :: "Player" assumes "v \<in> VV p" shows "v \<notin> VV p**"
not_flat_ang_def_equiv: fixes a :: "'p \<Rightarrow> 'p \<Rightarrow> 'p \<Rightarrow> bool" shows "(QCongAnFlat a) = (QCongA a \<and> (\<exists>A B C. a A B C \<and> \<not> Bet A B C))"
p_def: shows "p \<equiv> Polynomial.smult c (\<Prod>x\<in>#A. [:- x, 1::'a:])"
atU_empty: fixes U :: "'f set" shows "\<not> atU U {#}"
discounted_init: fixes r :: "real"   and X :: "nat \<Rightarrow> 'c \<Rightarrow> real" shows "\<forall>w. discounted_value r X 0 w = X 0 w"
cupcake_pmatch_preserve: fixes pat :: "pat"   and v0 :: "v"   and env :: "(char list \<times> v) list"   and envC :: "(char list, char list, nat \<times> tid_or_exn) namespace"   and env' :: "(char list \<times> v) list" assumes "is_cupcake_pat pat"   and "is_cupcake_value v0"   and "list_all (is_cupcake_value \<circ> snd) env"   and "cupcake_c_ns envC"   and "cupcake_pmatch envC pat v0 env = Match env'" shows "list_all (is_cupcake_value \<circ> snd) env'"
aff_dim_translation_eq: fixes a :: "'a"   and S :: "'a set" shows "aff_dim ((+) a ` S) = aff_dim S"
pow_eq_div2: fixes x :: "'a"   and m :: "nat"   and n :: "nat" assumes "x \<in> carrier G"   and "x [^] m = x [^] n" shows "x [^] (m - n) = \<one>"
sgn_cis: fixes a :: "real" shows "sgn (cis a) = cis a"
ortho_Snorm: shows "is_ortho_set (set Snorm)"
zip_eq_Nil_iff: fixes xs :: "'a list"   and ys :: "'b list" shows "(zip xs ys = []) = (xs = [] \<or> ys = [])"
subst_simp_var2: fixes x :: "'a"   and y :: "'a"   and M :: "'a trm" assumes "x \<noteq> y" shows "Var x[y ::= M] = Var x"
mult_zero_sup_circ: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> y * bot)\<^sup>\<circ> = x\<^sup>\<circ> * (y * bot)\<^sup>\<circ>"
assms_5: fixes s1 :: "'s1" assumes "bad1 s1"   and "I1 s1" shows "callee_invariant_on oracle2 (\<lambda>s2. bad2 s2 \<and> X_bad s1 s2) \<I>"
B\<^sub>Y: shows "basis_in Y B\<^sub>Y"
tau_func: fixes M :: "('f, 'a, 'g) Module_scheme"   and N :: "('h, 'a, 'g) Module_scheme"   and P :: "('f \<times> 'h, 'a, 'g) Module_scheme" assumes "R module M"   and "R module N"   and "FM\<^bsub>R\<^esub> P M N" shows "tau R M N P \<in> M \<times>\<^sub>c N \<rightarrow> carrier P"
simps_2: fixes X :: "binding"   and \<Gamma> :: "binding list" shows "trm_dom (X # \<Gamma>) = vrs_of X \<union> trm_dom \<Gamma>"
rep_eq: fixes x :: "('c, 'd) lifted_pair"   and xa :: "('c, 'd) lifted_pair" shows "(x < xa) = ((\<forall>f. triple.pairs_less_eq (Rep_lifted_pair x f) (Rep_lifted_pair xa f)) \<and> \<not> (\<forall>f. triple.pairs_less_eq (Rep_lifted_pair xa f) (Rep_lifted_pair x f)))"
prim_sem_state_def: shows "prim_sem_state = fst prim_sem_res"
mem_Collect_eq: fixes a :: "'b"   and P :: "'b \<Rightarrow> bool" shows "(a \<in> rel_ext P) = P a"
maximal_ne: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "class.linorder less_eq less"   and "finite S"   and "S \<noteq> {}" shows "order.maximal less_eq f S \<noteq> {}"
simps: fixes eff :: "'d \<Rightarrow> 'e \<Rightarrow> 'e fset \<Rightarrow> bool"   and rules :: "'d stream"   and "structure" :: "'f set"   and sat :: "'f \<Rightarrow> 'e \<Rightarrow> bool"   and \<delta> :: "'e \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> ('g \<times> bool \<times> 'g) set"   and Ord :: "('h \<times> 'h) set"   and \<sigma> :: "'g \<times> 'f \<Rightarrow> 'h"   and pointsTo :: "'i \<Rightarrow> ('e, 'd, 'i) ctree"   and a :: "('e, 'd, 'i) ctree" assumes "Infinite_Soundness eff rules structure sat \<delta> Ord \<sigma>"   and "\<forall>l l'. pointsTo l \<noteq> Link l'" shows "Infinite_Soundness.cwf eff rules pointsTo a = ((\<exists>l. a = Link l \<and> Infinite_Soundness.cwf eff rules pointsTo (pointsTo l)) \<or> (\<exists>r s cts. a = cNode (s, r) cts \<and> r \<in> i.R rules \<and> eff r s (Infinite_Soundness.seqOf pointsTo |`| cts) \<and> (\<forall>x. x |\<in>| cts \<longrightarrow> Infinite_Soundness.cwf eff rules pointsTo x)))"
handle2: fixes ck :: "bool"   and env :: "v sem_env"   and s1 :: "'b state"   and e :: "exp0"   and s2 :: "'b state"   and v1 :: "v"   and pes :: "(pat \<times> exp0) list"   and bv :: "'b state \<times> (v, v) result" assumes "evaluate ck env s1 e (s2, Rerr (Rraise v1)) \<and> evaluate_match ck env s2 v1 pes v1 bv" shows "evaluate ck env s1 (Handle e pes) bv"
IH: fixes psx :: "char list list"   and x :: "char list list"   and y :: "char list list"   and z :: "cmd" assumes "ps' = px # psx"   and "(px, x, y, z) \<in> set procs"   and "containsCall procs z psx p'"   and "containsCall procs c ps p"   and "(p', ins, outs, c) \<in> set procs" shows "containsCall procs z (psx @ p' # ps) p"
IH: fixes y :: "nat"   and w :: "'person \<Rightarrow> 'color"   and p :: "'person" assumes "y < n"   and "local.valid w" shows "local.leaves y p (local.try_swap p' c\<^sub>1 c\<^sub>2 w) = local.leaves y p w"
AE_all_countable: fixes P :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and M :: "'c measure" shows "(AE x in M. \<forall>i. P i x) = (\<forall>i. almost_everywhere M (P i))"
morE2: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('f, 'd, 'e) F2"   and B1' :: "'g set"   and B2' :: "'h set"   and s1' :: "'g \<Rightarrow> ('f, 'g, 'h) F1"   and s2' :: "'h \<Rightarrow> ('f, 'g, 'h) F2"   and f :: "'d \<Rightarrow> 'g"   and g :: "'e \<Rightarrow> 'h"   and z :: "'e" assumes "mor B1 B2 s1 s2 B1' B2' s1' s2' f g"   and "z \<in> B2" shows "F2map id f g (s2 z) = s2' (g z)"
drop: fixes v :: "v" shows "\<lparr>[$C v, $Drop]\<rparr> \<leadsto> \<lparr>[]\<rparr>"
val_Bool_def: fixes i :: "tr" shows "\<lbrakk>i\<rbrakk> = (THE j. i = Def j)"
is_sentence_b1_u: shows "is_sentence b1 \<and> is_sentence u"
Var: shows "listsp (\<lambda>x. WN x \<and> (\<exists>r'. x \<rightarrow>\<^sub>l r')) rs"
progress_le_gen: fixes j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and \<phi> :: "Formula.formula" assumes "pred_mapping (\<lambda>x. x \<le> j) P" shows "Monitor.progress \<sigma> P \<phi> j \<le> j"
edge_step: fixes a :: "'form"   and b :: "nat list"   and ba :: "('form, 'var) out_port"   and aa :: "'form"   and bb :: "nat list"   and bc :: "('form, 'var) antecedent" assumes "(((a, b), ba), (aa, bb), bc) \<in> edges" obtains i :: "nat" where "a = aa" and "b = bb @ [ia__]" and "bc = in_port_at (aa, bb) ia__" and "hyps (nodeOf (a, b)) ba = None"   | i :: "nat" where "a = aa" and "prefix (b @ [i]) bb" and "hyps (nodeOf (a, b)) ba = Some (in_port_at (a, b) i)"
mod_periodic_arithmetic: fixes f :: "nat \<Rightarrow> complex"   and k :: "nat"   and n :: "nat"   and m :: "nat" assumes "periodic_arithmetic f k"   and "n mod k = m mod k" shows "f n = f m"
shows_list_list_def: fixes xss :: "'a list list" shows "shows_list xss = showsp_list shows_prec 0 xss"
absorb1: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b" shows "min a b = a"
flopped_half_chamber_systems_gf: shows "folding_g.\<C> - g \<turnstile> folding_g.\<C> = f \<turnstile> folding_g.\<C>"
splitDigit_def: fixes p :: "'d \<Rightarrow> bool"   and i :: "'d"   and d :: "('c, 'd) Digit" shows "splitDigit p i d = splitNlist p i (digitToNlist d)"
abs_eq: fixes x :: "32 word" shows "uint_of_uint32 (Abs_uint32 x) = Abs_uint (ucast x)"
insert_before_is_component_unsafe: fixes h :: "(_) heap"   and ptr' :: "(_) object_ptr"   and child :: "(_) node_ptr"   and ref :: "(_) node_ptr option"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr"   and getter :: "(_) RObject_ext \<Rightarrow> 'm" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> insert_before ptr' child ref \<rightarrow>\<^sub>h h'"   and "ptr \<notin> set |h \<turnstile> get_dom_component ptr'|\<^sub>r"   and "ptr \<notin> set |h \<turnstile> get_dom_component (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child)|\<^sub>r"   and "ptr \<notin> set |h \<turnstile> get_dom_component (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r |h \<turnstile> get_owner_document ptr'|\<^sub>r)|\<^sub>r"   and "ptr \<notin> set |h \<turnstile> get_dom_component (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r |h \<turnstile> get_owner_document (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child)|\<^sub>r)|\<^sub>r" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"
inf_llist_inj: fixes f :: "nat \<Rightarrow> 'c"   and g :: "nat \<Rightarrow> 'c" shows "(inf_llist f = inf_llist g) = (f = g)"
mset_list_of_mset: fixes m :: "'a multiset" shows "mset (list_of_mset m) = m"
silent_tl: fixes s :: "'x \<times> 'm"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and s' :: "'x \<times> 'm" assumes "\<tau>move s ta s'" shows "ta = \<lbrace>\<rbrace>"
rel_option_iff: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a option"   and y :: "'b option" shows "rel_option R x y = (case (x, y) of (None, None) \<Rightarrow> True | (None, Some a) \<Rightarrow> False | (Some x, None) \<Rightarrow> False | (Some x, Some y) \<Rightarrow> R x y)"
rcs_in_Gp: fixes H :: "'a set"   and K :: "'a set"   and a :: "'a" assumes "G \<guillemotright> H "   and "G \<guillemotright> K "   and "K \<subseteq> H"   and "a \<in> H" shows "K \<bullet> a = K \<bullet>\<^bsub>\<natural>H\<^esub> a"
star_seq2: fixes c1 :: "com"   and s :: "char list \<Rightarrow> int"   and c1' :: "com"   and s' :: "char list \<Rightarrow> int"   and c2 :: "com" assumes "(c1, s) \<rightarrow>* (c1', s')" shows "(c1;; c2, s) \<rightarrow>* (c1';; c2, s')"
Smith_final: shows "Smith_mxn A = (P3' * P_H2 * P2' * P1', S, Q1 * Q2 * Q3')"
star_plus_loops: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "x\<^sup>\<odot> \<oplus> mone = matrix_kleene_algebra.tc x \<oplus> mone"
validI: fixes \<Gamma> :: "'g \<Rightarrow> ('e, 'g, 'f) com option"   and c :: "('e, 'g, 'f) com"   and P :: "'e set"   and F :: "'f set"   and Q :: "'e set"   and A :: "'e set" assumes "\<And>s t. \<lbrakk>\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow> t; s \<in> P; t \<notin> Fault ` F\<rbrakk> \<Longrightarrow> t \<in> Normal ` Q \<union> Abrupt ` A" shows "\<Gamma>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A"
elements_matrix_distribution_1: fixes A1 :: "'a list list"   and A2 :: "'a list list"   and B1 :: "'a list list"   and B2 :: "'a list list"   and i :: "nat"   and j :: "nat" assumes "mat (row_length A1) (length A1) A1"   and "mat (row_length A2) (length A2) A2"   and "mat (row_length B1) (length B1) B1"   and "mat (row_length B2) (length B2) B2"   and "length A1 = row_length A2"   and "length B1 = row_length B2"   and "A1 \<noteq> [] \<and> A2 \<noteq> [] \<and> B1 \<noteq> [] \<and> B2 \<noteq> []"   and "i < row_length A1 * row_length B1"   and "j < length A2 * length B2" shows "(A1 \<circ> A2 \<otimes> B1 \<circ> B2) ! j ! i = scalar_product (row A1 (i div row_length B1)) (col A2 (j div length B2)) * scalar_product (row B1 (i mod row_length B1)) (col B2 (j mod length B2))"
1: shows "- 1 < \<alpha>' \<and> \<alpha>' \<le> 1"
pgallery_chamber_system: fixes Cs :: "'a set list" assumes "pgallery Cs" shows "order.greater_eq folding_g.\<C> (set Cs)"
setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p_subst': fixes a :: "'f strand_label \<times> ('e, 'd) stateful_strand_step"   and \<theta> :: "'d \<Rightarrow> ('e, 'd) Term.term" assumes "set (bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p (snd a)) \<inter> subst_domain \<theta> = {}" shows "setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p (a \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>) = (\<lambda>(i, p). (i, p \<cdot>\<^sub>p \<theta>)) ` setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p a"
U_def: shows "U = {(i, n). 0 < i \<and> i \<le> n}"
simps_1: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and Doc :: "character list"   and k :: "nat" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel" shows "LocalLexing.\<T> \<TT> \<RR> \<SS> Lex Sel Doc k 0 = {}"
lift_def: shows "lift = rewrite_prog (\<lambda>_. lift_instr)"
is_unit_polyE': fixes p :: "'a poly" assumes "is_unit p" obtains a :: "'a" where "p = monom a 0" and "a \<noteq> (0::'a)"
normal_form_ideal_Polys: fixes X :: "'x set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite X"   and "F \<subseteq> P[X]" shows "normal_form (ideal F \<inter> P[X]) = normal_form F"
stirling_integral_bound: fixes n :: "nat" assumes "0 < n" obtains c :: "real" where "\<And>s. 0 < Re s \<Longrightarrow> cmod (stirling_integral n s) \<le> c / Re s ^ n"
subst_typ_rename_tvar_preserves_eq: fixes y :: "variable"   and S :: "String.literal set"   and T :: "typ"   and \<tau> :: "typ"   and x :: "variable" assumes "(y, S) \<notin> tvsT T"   and "(y, S) \<notin> tvsT \<tau>"   and "subst_typ [((x, S), Tv y S)] T = subst_typ [((x, S), Tv y S)] \<tau>" shows "T = \<tau>"
Smith_normal_form_upt_k_imp_Suc_k: fixes bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat" assumes "is_bezout_ext bezout"   and "Smith_normal_form_upt_k (diagonal_to_Smith_aux A [0..<k] bezout) k"   and "k < min (nrows A) (ncols A)" shows "Smith_normal_form_upt_k (diagonal_to_Smith_aux A [0..<Suc k] bezout) k"
q2_sat_props: shows "1 < length qs \<longrightarrow> p \<noteq> 0 \<and> 0 < length qs \<and> (\<forall>q. List.member qs q \<longrightarrow> coprime p q) \<longrightarrow> satisfies_properties p (drop (length qs div 2) qs) (get_subsets (calculate_data p (drop (length qs div 2) qs))) (get_signs (calculate_data p (drop (length qs div 2) qs))) (get_matrix (calculate_data p (drop (length qs div 2) qs)))"
hull_rel_relcomp_subset: fixes R :: "('a \<times> 'b) set" assumes "eqvt R" shows "R O hull_rel \<subseteq> hull_rel O R"
fps_compose_power: fixes c :: "'a fps"   and a :: "'a fps"   and n :: "nat" assumes "c $ 0 = (0::'a)" shows "(a oo c) ^ n = a ^ n oo c"