hcju/isabelleps huggingface dataset:
isTrue_iff_bprv_PPf: fixes var :: "'d set"   and trm :: "'e set"   and fmla :: "'f set"   and Var :: "'d \<Rightarrow> 'e"   and FvarsT :: "'e \<Rightarrow> 'd set"   and substT :: "'e \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> 'e"   and Fvars :: "'f \<Rightarrow> 'd set"   and subst :: "'f \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> 'f"   and eql :: "'e \<Rightarrow> 'e \<Rightarrow> 'f"   and cnj :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and imp :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and all :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and exi :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and fls :: "'f"   and dsj :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and num :: "'e set"   and enc :: "'f \<Rightarrow> 'e"   and prv :: "'f \<Rightarrow> bool"   and bprv :: "'f \<Rightarrow> bool"   and P :: "'f"   and isTrue :: "'f \<Rightarrow> bool"   and Pf :: "'f"   and n :: "'e"   and \<phi> :: "'f" assumes "Minimal_Truth_Soundness_HBL1iff_Compl_Pf var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num enc prv bprv P isTrue Pf"   and "n \<in> num"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "isTrue (Minimal_Truth_Soundness_HBL1iff_Compl_Pf.PPf var Var FvarsT Fvars subst Pf n (enc \<phi>)) = bprv (Minimal_Truth_Soundness_HBL1iff_Compl_Pf.PPf var Var FvarsT Fvars subst Pf n (enc \<phi>))"
tendsto_const: fixes "open" :: "'e set \<Rightarrow> bool"   and k :: "'e"   and F :: "'f filter" assumes "class.topological_space open" shows "LIM x F. k :> topological_space.nhds open k"
simps_1: fixes t1 :: "'c Tree" shows "merge t1 E = t1"
semilat_le_err_OK1: fixes x :: "'c"   and A :: "'c set"   and y :: "'c"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c err \<Rightarrow> 'c err \<Rightarrow> 'c err"   and z :: "'c" assumes "x \<in> A"   and "y \<in> A"   and "semilat (err A, Err.le r, f)"   and "OK x \<squnion>\<^bsub>f\<^esub> OK y = OK z" shows "x \<sqsubseteq>\<^bsub>r\<^esub> z"
simps_2: fixes \<beta> :: "'c \<Rightarrow> 'd \<Rightarrow> ternaryvalue"   and p :: "'d"   and m :: "'c match_expr" shows "map_match_tac \<beta> p (MatchNot m) = TernaryNot (map_match_tac \<beta> p m)"
not_less_zero: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and n :: "'a" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "\<not> less n zero"
l_distr: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x + y) * z = x * z + y * z"
fds_eval_shift: fixes c :: "'a"   and f :: "'a fds"   and z :: "'a" shows "eval_fds (fds_shift c f) z = eval_fds f (z - c)"
optim_def: shows "optim \<equiv> rewrite_prog optim_instr"
Q4''_a_def: shows "Q4''_a \<equiv> [: p, t, lnk, lbl, mrk \<leadsto> p', t', lnk', lbl', mrk' . p \<noteq> nil \<and> (\<forall>i. \<not> g mrk lbl (lnk i) p) \<and> t \<noteq> nil \<and> p' = t \<and> t' = lnk (lbl t) t \<and> lnk' = lnk(lbl t := (lnk (lbl t))(t := p)) \<and> lbl' = lbl(t := none) \<and> mrk' = mrk :]"
rev_induct2: fixes xs :: "'a list"   and ys :: "'b list"   and P :: "'a list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "length xs = length ys"   and "P [] []"   and "\<And>x xs y ys. \<lbrakk>length xs = length ys; P xs ys\<rbrakk> \<Longrightarrow> P (xs @ [x]) (ys @ [y])" shows "P xs ys"
prems_2: shows "(l # ts) ! i = (p, is, xs, Write\<^sub>s\<^sub>b False a sop v A L R W # sb, \<D>, \<O>, \<R>)"
tarski_prod: fixes x :: "'a"   and y :: "'a" assumes "\<And>x. x \<sqinter> nc \<noteq> (0::'a) \<Longrightarrow> nc \<cdot> ((x \<sqinter> nc) \<cdot> nc) = nc"   and "\<And>x y z. d x \<cdot> (y \<cdot> z) = d x \<cdot> y \<cdot> z" shows "(x \<sqinter> nc) \<cdot> nc \<cdot> ((y \<sqinter> nc) \<cdot> nc) = (if y \<sqinter> nc = (0::'a) then 0::'a else (x \<sqinter> nc) \<cdot> nc)"
chamber_vertex_conv_special_coset: fixes S :: "'b set"   and w :: "'b"   and X :: "'b set" assumes "CoxeterComplex S"   and "w \<in> \<langle>S\<rangle>"   and "X \<in> dual_order.poset_simplex_map (PreCoxeterSystem.special_cosets S) {w}" shows "\<exists>s\<in>S. X = w +o \<langle>S - {s}\<rangle>"
wt_elim_cases_30: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and C :: "qtname"   and x :: "tys" assumes "E,dt\<Turnstile>In1r (Init C)\<Colon>x" obtains "x = Inl (PrimT Void)" and "is_class (prg E) C"
prob_space_bind: fixes M :: "'c measure"   and N :: "'c \<Rightarrow> 'd measure"   and S :: "'d measure" assumes "prob_space M"   and "AE x in M. prob_space (N x)"   and "N \<in> M \<rightarrow>\<^sub>M subprob_algebra S" shows "prob_space (M \<bind> N)"
index_of_aux_more: fixes n :: "nat"   and ls :: "'a list"   and p :: "'a" shows "n \<le> index_of_aux ls p n"
assms_4: shows "wf\<^sub>s\<^sub>t\<^sub>s' \<S> (decomp_rm\<^sub>e\<^sub>s\<^sub>t \<A>\<^sub>d)"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>A\<^sub>1\<^sub>x\<^sub>A\<^sub>2) (\<cdot>\<^sub>B) F G \<tau>"
s2p_union_hom: fixes A :: "'b set"   and B :: "'b set" shows "s2p (A \<union> B) = s2p A \<squnion> s2p B"
Extend_empty_satisfies0: fixes \<AA> :: "'i"   and P :: "'v"   and k :: "'k"   and i :: "nat"   and a :: "'a" assumes "Length \<AA> = 0"   and "len P = 0" shows "(Extend k i \<AA> P \<Turnstile>\<^sub>0 a) = (\<AA> \<Turnstile>\<^sub>0 a)"
loxodromic_attracting_fixed_point_1: fixes f :: "'b \<Rightarrow> 'b" assumes "loxodromic_isometry f" shows "attracting_fixed_point f \<in> Gromov_boundary"
simps: fixes a :: "letter list list" shows "bar a = ((\<exists>ws. a = ws \<and> good ws) \<or> (\<exists>ws. a = ws \<and> (\<forall>x. bar (x # ws))))"
le_supI: fixes x :: "'c"   and a :: "'c"   and b :: "'c" assumes "x \<le> a"   and "x \<le> b" shows "x \<le> a \<sqinter> b"
squarefree_mono: fixes a :: "'b"   and b :: "'b" assumes "a dvd b"   and "squarefree b" shows "squarefree a"
lookup_raw_update_by_fun_raw: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and xs :: "('b \<times> 'c) list \<times> 'a"   and k1 :: "'b"   and f :: "'c \<Rightarrow> 'c"   and k2 :: "'b" assumes "oalist_raw.oalist_inv rep_key_order xs" shows "oalist_raw.lookup_raw rep_key_order (oalist_raw.update_by_fun_raw rep_key_order k1 f xs) k2 = (if k1 = k2 then f else id) (oalist_raw.lookup_raw rep_key_order xs k2)"
PLUS_def: fixes xs :: "'a language list" shows "PLUS xs \<equiv> foldr Plus xs Zero"
wlp_complete: fixes dims :: "nat list"   and S :: "com"   and P :: "complex mat" assumes "state_sig.well_com dims S"   and "state_sig.is_quantum_predicate dims P" shows "state_sig.hoare_partial dims (state_sig.wlp dims S P) S P"
mark_step: fixes prb :: "('d, 'e, 'f) pre_RedBlack"   and rv :: "'d \<times> nat"   and prb' :: "('d, 'e, 'f) pre_RedBlack" assumes "RedBlack prb"   and "mark_extends prb rv prb'" shows "RedBlack prb'"
BA: fixes i :: "'n" shows "(B *v y) $h i \<le> (A *v y) $h i"
ts__ncol: fixes A :: "'p"   and B :: "'p"   and X :: "'p"   and Y :: "'p" assumes "A B TS X Y" shows "\<not> Col A X Y \<or> \<not> Col B X Y"
bounded_measure: fixes M :: "'a measure"   and A :: "'a set" assumes "finite_measure M" shows "Sigma_Algebra.measure M A \<le> Sigma_Algebra.measure M (space M)"
face_of_convex_hull_insert_eq: fixes S :: "'b set"   and a :: "'b"   and F :: "'b set" assumes "finite S"   and "a \<notin> affine hull S" shows "(F face_of convex hull insert a S) = (F face_of convex hull S \<or> (\<exists>F'. F' face_of convex hull S \<and> F = convex hull insert a F'))"
bij_betw_prime_powers: fixes n :: "nat" shows "bij_betw (\<lambda>g. \<Prod>p\<in>{p. p \<le> n \<and> prime p}. p ^ g p) ({p. p \<le> n \<and> prime p} \<rightarrow>\<^sub>E UNIV) {m. 0 < m \<and> prime_factors m \<subseteq> {..n}}"
le_inf_iff: fixes x :: "'k multiset"   and y :: "'k multiset"   and z :: "'k multiset" shows "(x \<subseteq># y \<inter># z) = (x \<subseteq># y \<and> x \<subseteq># z)"
plus_left_distributivity: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a * (b + c) = a * b + (a * c)"
Power_right_lang: fixes M :: "'b nfa"   and qs :: "hf" assumes "nfa M"   and "qs \<in> dfa.states (nfa.Power_dfa M)" shows "dfa.right_lang (nfa.Power_dfa M) qs = \<Union> (nfa.right_lang M ` hfset qs)"
fvt: fixes t :: "'trm" assumes "set (y # map snd txs) \<subseteq> var"   and "set (s # map fst txs) \<subseteq> trm"   and "set [\<phi>] \<subseteq> fmla"   and "t \<in> set (s # map fst txs)" shows "set (getFrN (y # map snd txs) (s # map fst txs) [\<phi>] (length txs)) \<inter> FvarsT t = {}"
nres_cases: fixes M :: "'c nres" obtains "M = FAIL"   | X :: "'c set" where "M = RES X"
flowsto_union: fixes X0 :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set"   and T :: "real set"   and CX :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set"   and Y :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set"   and Z :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set"   and S :: "real set"   and CZ :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set"   and W :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set" assumes "flowsto X0 T CX Y"   and "flowsto Z S CZ W" shows "flowsto (X0 \<union> Z) (T \<union> S) (CX \<union> CZ) (Y \<union> W)"
distinct_27: shows "cospan_shape.TT \<noteq> cospan_shape.BT"
None_notin_image_Some: fixes A :: "'d set" shows "None \<notin> Some ` A"
isSPath_nt_parallel: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and s :: "nat"   and p :: "(nat \<times> nat) list"   and t :: "nat"   and e :: "nat \<times> nat" assumes "Graph.isSimplePath c s p t"   and "e \<in> set p" shows "prod.swap e \<notin> set p"
merge_status_eq_iff_1: fixes a :: "status" shows "(merge_status a SUCCESS = SUCCESS) = (a = SUCCESS)"
assoc_simps_3: fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h" shows "trg\<^sub>D \<a>\<^sub>D[f, g, h] = trg\<^sub>D f"
abs_nat_nat_def: shows "abs_nat = (\<lambda>x. x)"
integral_nonneg_AE: fixes f :: "'d \<Rightarrow> real"   and M :: "'d measure" assumes "AE x in M. 0 \<le> f x" shows "0 \<le> integral\<^sup>L M f"
composable: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data"   and \<nu> :: "'b arrow_of_spans_data" assumes "two_composable_arrows_of_spans C prj0 prj1 \<mu> \<nu>" shows "span_vertical_category.src C \<mu> = span_vertical_category.trg C \<nu>"
lns: shows "\<forall>x\<in>carrier. \<forall>e>0. \<exists>y\<in>carrier. norm (y - x) \<le> e \<and> u x < u y"
simps: fixes xs :: "('a \<times> 'c) list"   and ko :: "'o" shows "tl_raw (xs, ko) = (list.tl xs, ko)"
S_def: shows "S \<equiv> \<lambda>n i. fps_const (real (Stirling n i))"
arctan_inverse_neg: fixes x :: "real" assumes "x < 0" shows "arctan x = - pi / 2 - arctan (inverse x)"
Sigma_mset_def: fixes A :: "'a multiset"   and B :: "'a \<Rightarrow> 'b multiset" shows "Sigma_mset A B \<equiv> \<Sum>a\<in>#A. image_mset (Pair a) (B a)"
Cons_3: shows "(a # t') @ u =\<^sub>F v @ w"
g_imp_f: fixes i :: "real" assumes "g i = 0" shows "f i \<noteq> 0"
continuous_map_closure_preimage_subset: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and T :: "'d set" assumes "continuous_map X Y f" shows "X closure_of {x \<in> topspace X. f x \<in> T} \<subseteq> {x \<in> topspace X. f x \<in> Y closure_of T}"
bij_swap_iff: fixes a :: "'b"   and b :: "'b"   and f :: "'b \<Rightarrow> 'c" shows "bij (Fun.swap a b f) = bij f"
cospoly_Suc: fixes n :: "nat" shows "cospoly (Suc n) = (\<lambda>x. cospoly n x + cos_coeff n * x ^ n)"
tminus_left_distrib: fixes a :: "real"   and b :: "real"   and c :: "real" assumes "0 \<le> a" shows "a * (b \<ominus> c) = a * b \<ominus> a * c"
prim_comm_exp: fixes v :: "'b list"   and u :: "'b list" assumes "primitive v"   and "v \<cdot> u = u \<cdot> v" obtains k :: "nat" where "u = v \<^sup>@ k"
set_annul_def: fixes b :: "bool" shows "set_annul b \<equiv> do modify (annul_mod b); return () od"
add_diff_assoc: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a \<le> b" shows "c + (b - a) = c + b - a"
a_rcos_module_minus: fixes x :: "'a"   and x' :: "'a" assumes "ring R"   and "x \<in> carrier R"   and "x' \<in> carrier R" shows "(x' \<in> I +> x) = (x' \<ominus> x \<in> I)"
is_run_def: fixes r :: "nat \<Rightarrow> 'a" shows "is_run r \<equiv> r 0 \<in> V0 \<and> ipath E r"
npMulExp: fixes x :: "'a"   and n :: "nat"   and m :: "nat" assumes "x \<in> carrier R" shows "x^\<^bsup>R n\<^esup>^\<^bsup>R m\<^esup> = x^\<^bsup>R (n * m)\<^esup>"
pairwise_sdropD: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a stream"   and i :: "nat" assumes "PTA_Reachability.pairwise P xs" shows "PTA_Reachability.pairwise P (sdrop i xs)"
right_idem: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" shows "linordered_monoid.max (linordered_monoid.max a b) b = linordered_monoid.max a b"
trans_onode_comp: fixes i :: "nat"   and S :: "((nat \<Rightarrow> 'd) \<times> 'e, 'f seq_action) automaton"   and R :: "nat set" shows "automaton.trans (\<langle>i : S : R\<rangle>\<^sub>o) = onode_sos (automaton.trans S)"
induces_limit_situation: shows "limit_situation (\<cdot>\<^sub>J) (\<cdot>) D replete_setcat.comp \<phi> \<iota> a \<Phi> \<chi>"
Port_Graph_axioms: shows "Port_Graph nodes inPorts outPorts vertices nodeOf edges"
greatest: fixes j :: "nat" assumes "k cd\<^bsup>\<pi>\<^esup>\<rightarrow> j"   and "ipd (\<pi> j) \<noteq> ipd (\<pi> k)" shows "j \<le> (GREATEST j. k cd\<^bsup>\<pi>\<^esup>\<rightarrow> j \<and> ipd (\<pi> j) \<noteq> ipd (\<pi> k))"
Semi: fixes c1 :: "com"   and c2 :: "com"   and cs :: "com list"   and s :: "state" shows "((c1; c2) # cs, s) \<rightarrow> (c1 # c2 # cs, s)"
abs_eq: fixes x :: "(char list \<times> 'a) list" shows "fun (St x) = fun_rep x"
hf_equalityE: fixes A :: "hf"   and B :: "hf" assumes "A = B" obtains "A \<le> B" and "B \<le> A"
Throw_lemma: fixes tn :: "qtname"   and G :: "prog"   and x1 :: "abopt"   and s1 :: "st"   and L :: "lenv"   and a' :: "val" assumes "G\<turnstile>tn\<preceq>\<^sub>C SXcpt Throwable"   and "wf_prog G"   and "(x1, s1)\<Colon>\<preceq>(G, L)"   and "x1 = None \<longrightarrow> G,s1\<turnstile>a'\<Colon>\<preceq>Class tn" shows "(throw a' x1, s1)\<Colon>\<preceq>(G, L)"
assms: shows "\<not> eFreshAbs MOD ys y eA"
cl_subset_carrier: fixes X :: "'a set" shows "\<E>.cl X \<subseteq> \<E>"
orthogonal_transformation_norm: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "orthogonal_transformation f" shows "norm (f x) = norm x"
degree_polyadd: fixes p :: "'b poly"   and n0 :: "nat"   and q :: "'b poly"   and n1 :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1" shows "Polynomial_Expression.degree (p +\<^sub>p q) \<le> max (Polynomial_Expression.degree p) (Polynomial_Expression.degree q)"
linear_injective_left_inverse: fixes f :: "('b, 'c) vec \<Rightarrow> ('b, 'd) vec" assumes "Vector_Spaces.linear (*s) (*s) f"   and "inj f" shows "\<exists>g. Vector_Spaces.linear (*s) (*s) g \<and> g \<circ> f = id"
rep_eq: fixes x :: "nat" shows "Rep_\<I> (point x) = (x, enat x)"
comm_ring_ow_def: fixes U :: "'a set"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "comm_ring_ow U plus zero minus uminus times \<equiv> comm_semiring_ow U plus times \<and> ab_group_add_ow U plus zero minus uminus"
hyps: assumes "infinite S" shows "n \<le> wellorder_class.enumerate S n"
map2_val_compat'D1: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and f :: "('b \<times> 'c) list \<times> 'a \<Rightarrow> ('b \<times> 'd) list \<times> 'a"   and zs :: "('b \<times> 'c) list \<times> 'a" assumes "oalist_raw.map2_val_compat' rep_key_order f"   and "oalist_raw.oalist_inv rep_key_order zs" shows "oalist_raw.oalist_inv rep_key_order (f zs)"
InstanceOf_\<tau>red1r_xt: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and T :: "ty" assumes "\<tau>red1gr uf P t h (e, xs) (e', xs')" shows "\<tau>red1gr uf P t h (e instanceof T, xs) (e' instanceof T, xs')"
sep_implE: fixes x :: "'a \<Rightarrow> bool"   and y :: "'a \<Rightarrow> bool"   and h :: "'a" assumes "(x \<longrightarrow>* y) h" obtains "\<forall>h'. h ## h' \<and> x h' \<longrightarrow> y (h + h')"
strict_boundedE: fixes a :: "'t"   and b :: "'t"   and c :: "'t" assumes "a \<prec>\<^sub>t ord_term_lin.min b c" obtains "a \<prec>\<^sub>t b" and "a \<prec>\<^sub>t c"
exhaust: fixes y :: "iface" obtains x :: "char list" where "y = Iface x"
sortedP_appendI: fixes x :: "real \<times> real"   and xs :: "(real \<times> real) list"   and ys :: "(real \<times> real) list" assumes "ccw'.sortedP x xs"   and "ccw'.sortedP x ys"   and "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set ys\<rbrakk> \<Longrightarrow> ccw' x x y" shows "ccw'.sortedP x (xs @ ys)"
unit_imp_dvd: fixes b :: "'b"   and a :: "'b" assumes "b dvd (1::'b)" shows "b dvd a"
dg_def: shows "\<DD> = [\<DD>\<lparr>Obj\<rparr>, \<DD>\<lparr>Arr\<rparr>, \<DD>\<lparr>Dom\<rparr>, \<DD>\<lparr>Cod\<rparr>]\<^sub>\<circ>"
nsqn\<^sub>r_lte_dsn: fixes dsn :: "nat"   and dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat" shows "nsqn\<^sub>r (dsn, dsk, flag, hops, nhip) \<le> dsn"
x_def: shows "x = set {\<langle>0\<^sub>\<nat>, x\<lparr>0\<^sub>\<nat>\<rparr>\<rangle>, \<langle>1\<^sub>\<nat>, x\<lparr>1\<^sub>\<nat>\<rparr>\<rangle>}"
pa_def: fixes p :: "fm" shows "pa p = qelim (prep p) cooper"
simps_30: fixes f1 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'd"   and f2 :: "nat \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> 'd"   and f3 :: "nat \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> 'd"   and f4 :: "'e \<Rightarrow> 'd"   and f5 :: "nat \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'd"   and x51 :: "nat"   and x52 :: "'e"   and x53 :: "'e" shows "(case RecvMarker x51 x52 x53 of Trans x xa xb \<Rightarrow> f1 x xa xb | Send x xa xb xc xd xe \<Rightarrow> f2 x xa xb xc xd xe | Recv x xa xb xc xd xe \<Rightarrow> f3 x xa xb xc xd xe | Snapshot x \<Rightarrow> f4 x | RecvMarker x xa xb \<Rightarrow> f5 x xa xb) = f5 x51 x52 x53"
null_measure_idem: fixes M :: "'c measure" shows "null_measure (null_measure M) = null_measure M"
comm_monoid_add_hom_0_def: fixes hom :: "'a \<Rightarrow> 'b" shows "comm_monoid_add_hom_0 hom \<equiv> monoid_add_hom_0 hom"
fun_type_inv_wf: fixes t :: "('fun, 'var) Term.term"   and f :: "'fun"   and T :: "('fun, 'atom) Term.term list" assumes "\<Gamma> t = Fun f T"   and "wf\<^sub>t\<^sub>r\<^sub>m t" shows "arity f = length T"
countable_Collect: fixes A :: "'c set"   and \<phi> :: "'c \<Rightarrow> bool" assumes "countable A" shows "countable {a \<in> A. \<phi> a}"
assm: shows "\<forall>X. f (\<Squnion>X) = \<Squnion>{f x |x. x \<in> X}"
in_star_iff_concat: fixes w :: "'b list"   and A :: "'b list set" shows "(w \<in> star A) = (\<exists>ws. set ws \<subseteq> A \<and> w = concat ws)"
get_dom_component_impl: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_dom_component :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and is_strongly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and is_weakly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_element_by_id :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and get_elements_by_tag_name :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr list) prog" assumes "l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_tag_name" shows "get_dom_component = l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_dom_component get_root_node to_tree_order"
strict_monoidal_functor_axioms: shows "strict_monoidal_functor (\<cdot>) \<F>C.T\<^sub>F\<^sub>M\<^sub>C \<F>C.\<alpha>\<^sub>F\<^sub>M\<^sub>C \<F>C.\<iota>\<^sub>F\<^sub>M\<^sub>C (\<cdot>\<^sub>S) T\<^sub>F\<^sub>S\<^sub>M\<^sub>C \<alpha>\<^sub>F\<^sub>S\<^sub>M\<^sub>C \<iota>\<^sub>F\<^sub>S\<^sub>M\<^sub>C \<F>C.D"
ivp_3: fixes t :: "real"   and s :: "'a" assumes "t \<in> T"   and "s \<in> S" shows "(\<lambda>t. \<phi> t s) \<in> {0--t} \<rightarrow> S"
alt_def: fixes m1 :: "'m"   and m2 :: "'m" shows "alt m1 m2 = altc (cinsert \<^bold>1 (csingle \<^bold>2)) (\<lambda>c. if c = \<^bold>1 then m1 else m2)"
apath_def: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" shows "apath u p v \<equiv> awalk u p v \<and> distinct (pawalk_verts u p)"
local_lipschitz_minus: fixes T :: "'b set"   and X :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" shows "local_lipschitz T X (\<lambda>t x. - f t x) = local_lipschitz T X f"
assms_4: shows "(r, Expunge e2 mo r) \<in> set (node_deliver_messages xs)"
carrier_nxt_subset: fixes t :: "'a ttree"   and x :: "'a" shows "carrier (nxt t x) \<subseteq> carrier t"
assocs_self: fixes x :: "'a" assumes "x \<in> carrier G" shows "x \<in> assocs G x"
second_lesbegue_integral_eq: shows "LBINT x. LBINT y. f (y, x) = LBINT x. integral UNIV (\<lambda>y. f (y, x))"
finalVertexI: fixes g :: "graph"   and v :: "nat" assumes "Invariants.inv g"   and "final g"   and "v \<in> \<V> g" shows "finalVertex g v"
cr_fps_def: shows "cr_fps \<equiv> \<lambda>x y. x = ($) y"
Idl_subset_ideal: fixes I :: "'a set"   and H :: "'a set" assumes "ideal I R"   and "H \<subseteq> carrier R" shows "(Idl H \<subseteq> I) = (H \<subseteq> I)"
\<tau>move2_Seq1D: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and e1 :: "'addr expr1"   and e2 :: "'addr expr1"   and pc :: "nat"   and xcp :: "'addr option" assumes "\<tau>move2 P h stk (e1;; e2) pc xcp"   and "pc < length (compE2 e1)" shows "\<tau>move2 P h stk e1 pc xcp"
normalize_0: fixes x :: "'b" shows "(Projections.normalize x = (0::'b)) = (x = (0::'b))"
lunit_simps_3: fixes f :: "'a" assumes "B.ide f" shows "trg\<^sub>B (B.lunit f) = trg\<^sub>B f"
AllBallots_def: shows "AllBallots = \<Union> (range Ballot)"
height_balanced: fixes kdt :: "'b kdt" assumes "balanced kdt" shows "height kdt = nat \<lceil>log 2 (real (KD_Tree.size_kdt kdt))\<rceil>"
sym_Id_on: fixes A :: "'b set" shows "sym (Id_on A)"
max_n_nin_m: shows "Max_mset N \<notin># M"