hcju/isabelleps huggingface dataset:
neg_0_le_iff_le: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq zero (uminus a) = less_eq a zero"
simps: fixes a :: "nat" shows "ya_even a = (a = 0 \<or> (\<exists>x. a = x + 1 \<and> ya_odd x))"
comp_assoc: fixes f :: "'n \<Rightarrow> 'l"   and g :: "'m \<Rightarrow> 'n"   and h :: "'k \<Rightarrow> 'm" shows "f \<circ> g \<circ> h = f \<circ> (g \<circ> h)"
inf_aci_2: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.semilattice_inf inf less_eq less" shows "inf (inf x y) z = inf x (inf y z)"
AddSA_HAInitValue: fixes SA :: "('d, 'e, 'f) seqauto"   and HA :: "('d, 'e, 'f) hierauto"   and S :: "'d" assumes "States SA \<inter> HAStates HA = {}"   and "S \<in> HAStates HA" shows "HAInitValue (HA [++] (S, SA)) = HAInitValue HA"
increasing_def: fixes f :: "'d \<Rightarrow> 'e" shows "increasing f \<equiv> \<Inter>z. stable {s. z \<le> f s}"
eq_norm: fixes x :: "nat \<Rightarrow> bool" shows "(!!) (set_bits x) = (\<lambda>i. x i \<and> i < LENGTH('b))"
fls_base_factor_to_fps_nth: fixes f :: "'a fls"   and n :: "nat" shows "fls_base_factor_to_fps f $ n = fls_nth f (fls_subdegree f + int n)"
dg_prod_is_arrI: fixes f :: "V"   and a :: "V"   and b :: "V" assumes "vsv f"   and "\<D>\<^sub>\<circ> f = ZFC_in_HOL.set {j}"   and "vsv a"   and "\<D>\<^sub>\<circ> a = ZFC_in_HOL.set {j}"   and "vsv b"   and "\<D>\<^sub>\<circ> b = ZFC_in_HOL.set {j}"   and "\<And>i. i \<in>\<^sub>\<circ> ZFC_in_HOL.set {j} \<Longrightarrow> f\<lparr>i\<rparr> : a\<lparr>i\<rparr> \<mapsto>\<^bsub>\<CC>\<^esub> b\<lparr>i\<rparr>" shows "f : a \<mapsto>\<^bsub>\<Prod>\<^sub>D\<^sub>Gi\<in>\<^sub>\<circ>ZFC_in_HOL.set {j}. \<CC>\<^esub> b"
bisim1CASFail: fixes P :: "'b prog"   and h :: "'heap"   and e1 :: "'addr expr1"   and D :: "String.literal"   and F :: "String.literal"   and e2 :: "'addr expr1"   and e3 :: "'addr expr1"   and ad :: "'addr"   and xs :: "'addr val list"   and v' :: "'addr val"   and v :: "'addr val"   and v'' :: "'addr val" shows "P,e1\<bullet>compareAndSwap(D\<bullet>F, e2, e3),h \<turnstile> (Throw ad, xs) \<leftrightarrow> ([v', v, v''], xs, length (compE2 e1) + length (compE2 e2) + length (compE2 e3), \<lfloor>ad\<rfloor>)"
joinpaths_def: fixes g1 :: "real \<Rightarrow> 'c"   and g2 :: "real \<Rightarrow> 'c" shows "g1 +++ g2 = (\<lambda>x. if x \<le> 1 / 2 then g1 (2 * x) else g2 (2 * x - 1))"
arrE: fixes f :: "'a" assumes "hom'.arr f" shows "\<guillemotleft>f : trg e\<^sub>0 \<rightarrow> trg e\<^sub>1\<guillemotright>"
pp: assumes "w1 = POS_INF \<and> w2 = POS_INF" shows "P w1 w2"
stream_all_shift: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list"   and s :: "'b stream" shows "stream_all P (xs @- s) = (list_all P xs \<and> stream_all P s)"
lpp_e_focus: fixes g :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" assumes "g \<in> G" shows "i.lpp (poly_eval (\<lambda>_. monomial a 0) (focus {x} g)) = except (i.lpp g) {x}"
PPs_closed_minus: fixes s :: "'c \<Rightarrow>\<^sub>0 nat"   and X :: "'c set"   and t :: "'c \<Rightarrow>\<^sub>0 nat" assumes "s \<in> .[X]" shows "s - t \<in> .[X]"
swap_def: fixes a :: "'a \<Rightarrow> 'b"   and i :: "'a"   and j :: "'a" shows "swap a i j \<equiv> a(i := a j, j := a i)"
ext: fixes g :: "'a"   and f :: "'a" assumes "g \<cdot> f \<noteq> C.null" shows "C.seq g f"
smallstepV_consumes_proofstream: fixes \<pi>\<^sub>1 :: "Syntax.term list"   and eV :: "Syntax.term"   and \<pi>\<^sub>2 :: "Syntax.term list"   and eV' :: "Syntax.term" assumes "\<lless>\<pi>\<^sub>1, eV\<ggreater> V\<rightarrow> \<lless>\<pi>\<^sub>2, eV'\<ggreater>" obtains \<pi> :: "Syntax.term list" where "\<pi>\<^sub>1 = \<pi> @ \<pi>\<^sub>2"
leof_trans: fixes a :: "'c nres"   and X :: "'c set"   and c :: "'c nres" assumes "a \<le>\<^sub>n RES X"   and "RES X \<le>\<^sub>n c" shows "a \<le>\<^sub>n c"
neq_States: fixes SA :: "('d, 'e, 'f) seqauto" shows "States SA \<noteq> {}"
cmp_rule1: fixes s :: "'a"   and ni :: "'b"   and i :: "'c ifex"   and ni' :: "'b" assumes "I s"   and "(ni, i) \<in> R s"   and "(ni', i) \<in> R s" shows "cmp ni ni'"
Rep_cfun_inverse: fixes x :: "'c \<rightarrow> 'd" shows "Abs_cfun (Rep_cfun x) = x"
eventually_at_infinity: fixes P :: "'a \<Rightarrow> bool" shows "eventually P at_infinity = (\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> P x)"
strategy_attracts_via_v0: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and v0 :: "'a"   and A :: "'a set"   and W :: "'a set" assumes "strategy p \<sigma>"   and "strategy_attracts_via p \<sigma> v0 A W"   and "v0 \<in> V" shows "v0 \<in> A \<union> W"
simps_19: fixes x5 :: "char list" shows "Class x5 \<noteq> Integer"
fminus_partition: fixes A :: "'c fset"   and B :: "'c fset" assumes "A |\<subseteq>| B" shows "A |\<union>| (B |-| A) = B"
preT_transl: fixes C :: "Nielson_VCG.acom"   and E :: "(char list \<Rightarrow> int) \<Rightarrow> nat" shows "Nielson_VCGi.preT (transl C) E = Nielson_VCG.preT C E"
ENC_And: fixes n :: "nat"   and \<phi> :: "'a formula"   and \<psi> :: "'a formula" assumes "wf_formula n (FAnd \<phi> \<psi>)" shows "lang n (ENC n (FOV (FAnd \<phi> \<psi>))) - {[]} \<subseteq> lang n (ENC n (FOV \<phi>)) \<inter> lang n (ENC n (FOV \<psi>)) - {[]}"
f': fixes T :: "real set"   and X :: "'a set"   and f :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and f' :: "real \<times> 'a \<Rightarrow> real \<times> 'a \<Rightarrow> 'a"   and tx :: "real \<times> 'a" assumes "derivative_on_prod T X f f'"   and "tx \<in> T \<times> X" shows "((\<lambda>(t, x). f t x) has_derivative f' tx) (at tx within T \<times> X)"
simps_1: shows "exH_k 0 = hadamard_on_i 0"
sscan_scons: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'c"   and xs :: "'c stream"   and a :: "'b" shows "sscan f (x ## xs) a = f x a ## sscan f xs (f x a)"
faces: fixes X :: "'b set set"   and x :: "'b set"   and y :: "'b set" assumes "SimplicialComplex X"   and "x \<in> X"   and "order.greater_eq x y" shows "y \<in> X"
init_tableau_normalized: fixes t :: "(nat \<times> linear_poly) list" assumes "\<triangle> t" shows "\<triangle> (\<T> (init t))"
sub_def: fixes as :: "'c iarray"   and n :: "nat" shows "as !! n = IArray.list_of as ! n"
isInIDo: fixes p :: "('b, 'c) Combinators list"   and s :: "('b, 'c) Combinators" assumes "noDenyAll p"   and "s \<in> set (insertDenies p)" shows "\<exists>!a. s = DenyAllFromTo (first_srcNet a) (first_destNet a) \<oplus> DenyAllFromTo (first_destNet a) (first_srcNet a) \<oplus> a \<and> a \<in> set p"
crypts_hash_2: fixes X :: "msg" shows "insert (Hash X) (crypts {X}) \<subseteq> crypts {Hash X}"
get_ancestors_si_also_parent: fixes h :: "(_) heap"   and some_ptr :: "(_) object_ptr"   and ancestors :: "(_) object_ptr list"   and child :: "(_) node_ptr"   and parent :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "h \<turnstile> get_ancestors_si some_ptr \<rightarrow>\<^sub>r ancestors"   and "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 \<in> set ancestors"   and "h \<turnstile> get_parent child \<rightarrow>\<^sub>r Some parent"   and "type_wf h"   and "known_ptrs h" shows "parent \<in> set ancestors"
red_dfs_inv_initial: fixes E :: "('a \<times> 'a) set"   and V0 :: "'a set" assumes "finite (E\<^sup>* `` V0)" shows "red_dfs_inv E (E\<^sup>* `` V0) {} {}"
list_length: fixes s :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "length (list s n) = n"
lhd_LCons: fixes x21 :: "'b"   and x22 :: "'b llist" shows "lhd (LCons x21 x22) = x21"
mI_principal: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and I :: "'a set" assumes "0 < n"   and "distinct_pds K n P"   and "ideal (O\<^bsub>K P n\<^esub>) I"   and "I \<noteq> {\<zero>\<^bsub>O\<^bsub>K P n\<^esub>\<^esub>}"   and "I \<noteq> carrier (O\<^bsub>K P n\<^esub>)" shows "I = (O\<^bsub>K P n\<^esub>) \<diamondsuit>\<^sub>p (mIg\<^bsub> K n P I\<^esub>)"
compile1: shows "rec_ci (Cn n f gs) = (ap, ar, ft) \<and> length args = ar"
walk_iarray_foldl': fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and xs :: "'c list"   and x :: "'b" shows "walk_iarray f (IArray xs) x = foldl f x xs"
set_2: fixes x21 :: "'a tree"   and x22 :: "'a"   and x23 :: "'a tree" shows "set_tree \<langle>x21, x22, x23\<rangle> = set_tree x21 \<union> {x22} \<union> set_tree x23"
CompDefComp_2: fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "f \<approx>> g"   and "g \<approx>> h" shows "f ;; g \<approx>> h"
coeff_def: fixes p :: "'b mpoly" shows "coeff p = lookup (mapping_of p)"
queue_invar_simps_3: fixes q :: "('c, 'd) BinomialTree list"   and q' :: "('c, 'd) BinomialTree list" shows "queue_invar (q @ q') = (queue_invar q \<and> queue_invar q')"
lappend_inf: fixes xs :: "'a llist"   and ys :: "'a llist" assumes "\<not> lfinite xs" shows "xs @\<^sub>l ys = xs"
rbt_comp_insert_def: shows "local.rbt_comp_insert = local.rbt_comp_insert_with_key (\<lambda>_ _ nv. nv)"
invar_tail_invar: fixes t :: "('c, 'd) SkewBinomialTree"   and bq :: "('c, 'd) SkewBinomialTree list" assumes "invar (t # bq)" shows "tail_invar bq"
cprank_maxI: fixes BS :: "'a tensor list"   and ds :: "nat list" assumes "\<And>B. B \<in> set BS \<Longrightarrow> cprank_max1 B"   and "\<And>B. B \<in> set BS \<Longrightarrow> dims B = ds" shows "cprank_max (length BS) (listsum ds BS)"
real_v_integral_eq: fixes cfg :: "'s cfg" assumes "cfg \<in> valid_cfg" shows "enn2real (\<integral>\<^sup>+ s. v (cont cfg s) \<partial>measure_pmf (action cfg)) = measure_pmf.expectation (action cfg) (\<lambda>s. enn2real (v (cont cfg s)))"
left_total_MF_Rel: shows "left_total MF_Rel"
reachable_fun_upd: fixes s :: "'c process_name \<Rightarrow> ('b, 'c, 'd, 'a) Model.local_state"   and fld :: "'c process_name"   and val :: "('b, 'c, 'd, 'a) Model.local_state"   and m :: "'c"   and r :: "'a" assumes "roots ((s(fld := val)) (mutator m)) = roots (s (mutator m))"   and "ghost_honorary_root ((s(fld := val)) (mutator m)) = ghost_honorary_root (s (mutator m))"   and "\<And>x. \<Union> (ran ` obj_fields ` set_option (sys_heap (s(fld := val)) x)) = \<Union> (ran ` obj_fields ` set_option (sys_heap s x))"   and "filter (\<lambda>s. is_mw_Mutate s \<or> is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) (s(fld := val))) = filter (\<lambda>s. is_mw_Mutate s \<or> is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s)" shows "mut_m.reachable m r (s(fld := val)) = mut_m.reachable m r s"
set_iterator_filter_def: fixes P :: "'e \<Rightarrow> bool" shows "set_iterator_filter P \<equiv> set_iterator_image_filter (\<lambda>x. if P x then Some x else None)"
invar_empty_1: shows "SkewBinomialHeapStruc.invar []"
ftip_cons_botI: fixes cld :: "cld"   and fds :: "fd list"   and P :: "cld list"   and ctx :: "ctx"   and f :: "char list"   and ctxcld_list :: "(ctx \<times> cld) list" assumes "class_fields cld fds"   and "ftype_in_fds P ctx fds f ty_opt_bot_bot" shows "ftype_in_path P ([(ctx, cld)] @ ctxcld_list) f None"
triple_valid_def2: fixes n :: "nat"   and P :: "'b assn"   and c :: "com"   and Q :: "'b assn" shows "|=n:{P}. c .{Q} = (\<forall>Z s. P Z s \<longrightarrow> (\<forall>s'. <c,s> -n-> s' \<longrightarrow> Q Z s'))"
has_derivative_on_open2: fixes t :: "real"   and s :: "'a" assumes "t < 0"   and "t \<in> T"   and "s \<in> S" obtains B :: "real set" where "t \<in> B" and "open B" and "B \<subseteq> T" and "D (\<lambda>\<tau>. \<phi> \<tau> s) \<mapsto> (\<lambda>\<tau>. \<tau> *\<^sub>R f (\<phi> t s)) at t within B"
height_insert_ge: fixes t :: "'b tree"   and x :: "'b" shows "height t \<le> height (insert x t)"
crel_vs_iter_and_compute: fixes P :: "'a \<Rightarrow> bool"   and lookup :: "'b \<Rightarrow> ('a, 'c option) state"   and update :: "'b \<Rightarrow> 'c \<Rightarrow> ('a, unit) state"   and cnt :: "'b \<Rightarrow> bool"   and nxt :: "'b \<Rightarrow> 'b"   and sizef :: "'b \<Rightarrow> nat"   and dp :: "'b \<Rightarrow> 'c"   and R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and g :: "'b \<Rightarrow> 'd"   and f :: "'b \<Rightarrow> ('a, 'e) state"   and x :: "'b"   and y :: "'b" assumes "dp_consistency_iterator P lookup update cnt nxt sizef"   and "rel_fun (=) (dp_consistency.crel_vs lookup P dp R) g f" shows "dp_consistency.crel_vs lookup P dp R (g x) (iterator_defs.iter_state cnt nxt f y \<bind> (\<lambda>_. f x))"
util_def: fixes x :: "'a pmf"   and y :: "'a pmf" assumes "x \<in> \<P> outcomes"   and "y \<in> \<P> outcomes" shows "x \<succeq> y = (measure_pmf.expectation y u \<le> measure_pmf.expectation x u)"
fresh_int: fixes a :: "'a"   and i :: "int" shows "a \<sharp> i"
adopt_node_preserves_type_wf: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and child :: "(_) node_ptr"   and h' :: "(_) heap" assumes "heap_is_wellformed h"   and "h \<turnstile> adopt_node document_ptr child \<rightarrow>\<^sub>h h'"   and "known_ptrs h"   and "type_wf h" shows "type_wf h'"
subst_eq_dest: fixes s1 :: "('b \<times> 'b trm) list"   and s2 :: "('b \<times> 'b trm) list"   and t :: "'b trm" assumes "s1 \<doteq> s2" shows "t \<lhd> s1 = t \<lhd> s2"
finite_jumpFs_def: fixes f :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real" shows "finite_jumpFs f a b = finite {x. (jumpF f (at_left x) \<noteq> 0 \<or> jumpF f (at_right x) \<noteq> 0) \<and> a \<le> x \<and> x \<le> b}"
continuous_on_ereal: fixes s :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "continuous_on s f" shows "continuous_on s (\<lambda>x. ereal (f x))"
measurable_on_if_simple_function_limit: fixes g :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>n. g n measurable_on UNIV"   and "\<And>n. finite (range (g n))"   and "\<And>x. (\<lambda>n. g n x) \<longlonglongrightarrow> f x" shows "f measurable_on UNIV"
assms_1: shows "z \<sharp> [y].t\<^sub>1"
rep_smult: fixes c :: "'a \<Rightarrow>\<^sub>0 'c"   and r :: "('t \<Rightarrow>\<^sub>0 'c) \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'c" shows "pmdl.rep (monomial c 0 * r) = c \<odot> pmdl.rep r"
skip_joiner_def: shows "skip_joiner = (\<lambda>s. \<Union> (range s))"
S'_1: shows "set S' \<subseteq> {p} \<union> S"
ordinal_times_eq_0: fixes x :: "ordinal"   and y :: "ordinal" shows "(x * y = 0) = (x = 0 \<or> y = 0)"
z1pmod2: fixes b :: "int" shows "(2 * b + 1) mod 2 = 1"
joint_D_gchains: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and m :: "nat"   and g :: "nat \<Rightarrow> 'a set" assumes "D_gchain G n f"   and "D_gchain G m g"   and "g 0 \<subset> f n" shows "D_gchain G (Suc (n + m)) (jointfun n f m g)"
normalize_primitive_extract_preserves_nnf_normalized: fixes m :: "'b match_expr"   and disc :: "'b \<Rightarrow> bool"   and sel :: "'b \<Rightarrow> 'c"   and C :: "'c \<Rightarrow> 'b"   and f :: "'c negation_type list \<Rightarrow> 'c list" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C" shows "\<forall>mn\<in>set (normalize_primitive_extract (disc, sel) C f m). normalized_nnf_match mn"
assms_1: shows "\<forall>\<^sub>F x in at a. f x = g x"
init_le_def2: fixes G :: "prog"   and n :: "nat"   and s :: "state" shows "(G\<turnstile>init\<le>n) s = (card (nyinitcls G s) \<le> n)"
h2: shows "aEval (Eq p1) (L[var := sqrt Cv]) = aEval (Eq p2) (L[var := sqrt Cv])"
alwD: fixes P :: "'b stream \<Rightarrow> bool"   and x :: "'b stream" assumes "alw P x" shows "P x"
1: shows "r ; (x\<^sup>+ \<cdot> 1') ; 1 \<le> x\<^sup>T ; 1"
Keys_insert: fixes a :: "'c \<Rightarrow>\<^sub>0 'd"   and A :: "('c \<Rightarrow>\<^sub>0 'd) set" shows "Keys (insert a A) = keys a \<union> Keys A"
iso_hcomp: fixes \<mu> :: "'b"   and \<nu> :: "'b" assumes "local.iso \<mu>"   and "local.iso \<nu>"   and "src \<nu> = trg \<mu>" shows "local.iso (\<nu> \<star> \<mu>)"
left_nesting_neq_match: fixes f :: "term"   and g :: "'c" assumes "left_nesting f \<noteq> left_nesting g"   and "is_const (fst (strip_comb f))" shows "match f g = None"
run_altc_sample_option: fixes altc_sample :: "'c \<Rightarrow> ('d \<Rightarrow> 'a) \<Rightarrow> 'a"   and p :: "'c"   and f :: "'d \<Rightarrow> ('b, 'a) optionT" shows "run_option (altc_sample_option altc_sample p f) = altc_sample p (\<lambda>x. run_option (f x))"
welldefined_signs2: shows "well_def_signs (length qs2) signs2"
ideduct_synth_subst: fixes M :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term"   and \<delta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "M \<turnstile>\<^sub>c t" shows "M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<delta> \<turnstile>\<^sub>c t \<cdot> \<delta>"
lemma5: fixes e :: "Syntax.term"   and eP :: "Syntax.term"   and eV :: "Syntax.term"   and \<tau> :: "ty"   and e' :: "Syntax.term" assumes "{$$} \<turnstile> e, eP, eV : \<tau>"   and "\<lless>[], e\<ggreater> I\<rightarrow> \<lless>[], e'\<ggreater>" obtains eP' :: "Syntax.term" and eV' :: "Syntax.term" and \<pi> :: "Syntax.term list" where "{$$} \<turnstile> e', eP', eV' : \<tau>" and "\<forall>\<pi>\<^sub>P. \<lless>\<pi>\<^sub>P, eP\<ggreater> P\<rightarrow> \<lless>\<pi>\<^sub>P @ \<pi>, eP'\<ggreater>" and "\<forall>\<pi>'. \<lless>\<pi> @ \<pi>', eV\<ggreater> V\<rightarrow> \<lless>\<pi>', eV'\<ggreater>"
cases: fixes a1 :: "'c"   and a2 :: "'d node_action"   and a3 :: "'c"   and S :: "('c \<times> 'd node_action \<times> 'c) set" assumes "(a1, a2, a3) \<in> cnet_sos S" obtains s :: "'c" and i :: "nat" and i' :: "nat" and s' :: "'c" where "a1 = sb__" and "a2 = connect(ia__, i'__)" and "a3 = s'__" and "(sb__, connect(ia__, i'__), s'__) \<in> S"   | s :: "'c" and i :: "nat" and i' :: "nat" and s' :: "'c" where "a1 = sc__" and "a2 = disconnect(ib__, i')" and "a3 = s'a__" and "(sc__, disconnect(ib__, i'), s'a__) \<in> S"   | s :: "'c" and R :: "nat set" and m :: "'d" and s' :: "'c" where "a1 = sd__" and "a2 = \<tau>" and "a3 = s'b__" and "(sd__, R:*cast(m), s'b__) \<in> S"   | s :: "'c" and s' :: "'c" where "a1 = se__" and "a2 = \<tau>" and "a3 = s'c__" and "(se__, \<tau>, s'c__) \<in> S"   | s :: "'c" and i :: "nat" and d :: "nat" and s' :: "'c" where "a1 = sf__" and "a2 = ic__:deliver(d__)" and "a3 = s'd__" and "(sf__, ic__:deliver(d__), s'd__) \<in> S"   | s :: "'c" and i :: "nat" and K :: "nat set" and d :: "nat" and di :: "nat" and s' :: "'c" where "a1 = s" and "a2 = i:newpkt(d, di)" and "a3 = s'" and "(s, {i}\<not>K:arrive(newpkt (d, di)), s') \<in> S"
dvd_Lcm: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A" shows "a dvd Lcm A"
delete_shadow_root_pointers: fixes h :: "(_) heap"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and h' :: "(_) heap" assumes "h \<turnstile> delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr \<rightarrow>\<^sub>h h'" shows "object_ptr_kinds h = object_ptr_kinds h' |\<union>| {|cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^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 shadow_root_ptr|}"
star_slide_var1: fixes x :: "'a" shows "x\<^sup>\<star> ; x \<le> x\<^sup>+"
rel_setI: fixes A :: "'f set"   and B :: "'g set"   and R :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>B. R x y"   and "\<And>y. y \<in> B \<Longrightarrow> \<exists>x\<in>A. R x y" shows "rel_set R A B"
1: shows "\<forall>f. (- Rep_lifted_boolean_algebra xf f, Rep_phi f (Rep_lifted_boolean_algebra xf f)) \<in> triple.pairs (Rep_phi f) \<and> (Rep_lifted_boolean_algebra xf f, Rep_phi f (- Rep_lifted_boolean_algebra xf f)) \<in> triple.pairs (Rep_phi f)"
at_post_def: fixes x :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> \<langle>\<langle>'\<alpha>\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> \<upsilon> x" shows "\<tau> \<Turnstile> \<delta> H .allInstances()->includes\<^sub>S\<^sub>e\<^sub>t(x)"
prop_3_2_i: fixes X :: "'a set" shows "filterof X = {a. \<exists>n x. x \<in> X *^ n \<and> x \<le> a}"
Int_absorb: fixes A :: "'d set" shows "A \<inter> A = A"
simps: fixes s :: "('f, 'v) Term.term"   and t :: "('f, 'v) Term.term" shows "wpo s t = (case s of Var x \<Rightarrow> (False, case t of Var xa \<Rightarrow> x = xa | Fun g ts \<Rightarrow> (s, t) \<in> NS \<and> status \<sigma>\<sigma> (g, length ts) = [] \<and> prl (g, length ts)) | Fun f ss \<Rightarrow> if (s, t) \<in> S then (True, True) else if (s, t) \<in> NS then if \<exists>i\<in>set (status \<sigma>\<sigma> (f, length ss)). ss ! i \<succeq> t then (True, True) else case t of Var x \<Rightarrow> (False, ssimple \<and> large (f, length ss)) | Fun g ts \<Rightarrow> case prc (f, length ss) (g, length ts) of (prs, prns) \<Rightarrow> if prns \<and> (\<forall>j\<in>set (status \<sigma>\<sigma> (g, length ts)). s \<succ> ts ! j) then if prs then (True, True) else let ss' = map ((!) ss) (status \<sigma>\<sigma> (f, length ss)); ts' = map ((!) ts) (status \<sigma>\<sigma> (g, length ts)); cf = c (f, length ss); cg = c (g, length ts) in if cf = Lex \<and> cg = Lex then lex_ext wpo n ss' ts' else if cf = Mul \<and> cg = Mul then mul_ext wpo ss' ts' else (length ss' \<noteq> 0 \<and> length ts' = 0, length ts' = 0) else (False, False) else (False, False))"
image_mset_cong: fixes M :: "'k multiset"   and f :: "'k \<Rightarrow> 'l"   and g :: "'k \<Rightarrow> 'l" assumes "\<And>x. x \<in># M \<Longrightarrow> f x = g x" shows "image_mset f M = image_mset g M"
resl_comp2: fixes x :: "'a"   and y :: "'a" shows "(x \<leftarrow> y) \<cdot> y \<leftarrow> y = x \<leftarrow> y"
distinguished_bounded_support: fixes x :: "('idx, 'pred, 'act) formula"   and P :: "'state"   and Q :: "'state" assumes "weak_formula x"   and "x distinguishes P from Q" obtains y :: "('idx, 'pred, 'act) formula" where "weak_formula y" and "supp y \<subseteq> supp P" and "y distinguishes P from Q"
Node_def: shows "Old_Datatype.Node = {p. \<exists>f x k. p = (f, x) \<and> f k = Inr 0}"
HSem_def': fixes \<Gamma> :: "('c \<times> 'f) list"   and \<rho> :: "'c \<Rightarrow> 'd" shows "HSem2.HSem_syn \<Gamma> \<rho> = (\<mu> x. \<rho> ++\<^bsub>domA \<Gamma>\<^esub> HSem2.EvalHeapSem_syn \<Gamma> x)"
Inf_greatest: fixes A :: "'a set"   and z :: "'a" assumes "\<And>x. x \<in> A \<Longrightarrow> z \<le> x" shows "z \<le> Inf A"
assms_4: shows "proj2_incident (apply_cltn2 p C) m"
power_commuting_commutes: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c"   and n :: "nat" assumes "class.monoid_mult one times"   and "times x y = times y x" shows "times (power.power one times x n) y = times y (power.power one times x n)"
demonic_assert_comp: fixes r :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'a \<Rightarrow> bool" shows "[: r :] \<circ> {. p .} = {.x. \<forall>y. r x y \<longrightarrow> p y.} \<circ> [: r :]"
IH_1: fixes x :: "'a \<times> 'b"   and xa :: "('a \<times> 'b) set"   and xb :: "('a \<times> 'b) set" assumes "\<not> (infinite A \<or> A = {})"   and "x = arg_min_on snd A"   and "xa = {p \<in> A. fst p < fst x}"   and "xb = {p \<in> A. fst x < fst p}"   and "finite xa"   and "inj_on fst xa"   and "inj_on snd xa" shows "set_tree (treap_of xa) = xa"
b_in_L: shows "b \<in> H.face_cycle_set b"
det_minor_col: fixes B :: "('c, 'd) sq_matrix"   and i :: "'d"   and j :: "'d" shows "Square_Matrix.det (minor B i j) = Square_Matrix.det (upd_col B j (axis i (1::'c)))"
distinct_35: fixes pi :: "pi"   and name1' :: "name"   and name2' :: "name"   and pi' :: "pi" shows "\<tau>.pi \<noteq> name1'<name2'>.pi'"
isPath_edgeset: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat"   and e :: "nat \<times> nat" assumes "Graph.isPath c u p v"   and "e \<in> set p" shows "e \<in> Graph.E c"
ncol_conga_ncol: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "\<not> Col A B C"   and "A B C CongA D E F" shows "\<not> Col D E F"