hcju/isabelleps huggingface dataset:
Bseq_monoseq_convergent'_inc: fixes f :: "nat \<Rightarrow> real"   and M :: "nat" assumes "Bseq (\<lambda>n. f (n + M))"   and "\<And>m n. \<lbrakk>M \<le> m; m \<le> n\<rbrakk> \<Longrightarrow> f m \<le> f n" shows "convergent f"
drop_0: fixes xs :: "'g list" shows "drop 0 xs = xs"
someax: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula" shows "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H F \<^bold>\<rightarrow> \<^bold>\<not> F \<^bold>\<rightarrow> \<bottom>"
d_OUT_Inf: fixes Y :: "('a \<times> 'a \<Rightarrow> ennreal) set"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "Complete_Partial_Order.chain (\<le>) Y"   and "Y \<noteq> {}"   and "countable (support_flow f)"   and "\<And>g e. g \<in> Y \<Longrightarrow> g e \<le> f e"   and "d_OUT f x \<noteq> \<top>" shows "d_OUT (\<Sqinter> Y) x = (\<Sqinter>g\<in>Y. d_OUT g x)"
unique_declared_in: fixes G :: "prog"   and m :: "memberdecl"   and C :: "qtname"   and n :: "memberdecl" assumes "G\<turnstile> m declared_in C"   and "G\<turnstile> n declared_in C"   and "memberid m = memberid n" shows "m = n"
thetaParZO_def: shows "thetaParZO \<equiv> {(c1 | c2, d1 | d2) |c1 c2 d1 d2. c1 \<approx>01 d1 \<and> c2 \<approx>01 d2}"
reduced_bdt_checkers_axioms: shows "reduced_bdt_checkers ifex_of val"
bi_unique_conversep: fixes R :: "'b \<Rightarrow> 'a \<Rightarrow> bool" shows "bi_unique R\<inverse>\<inverse> = bi_unique R"
pmf_bind: fixes N :: "'b pmf"   and f :: "'b \<Rightarrow> 'a pmf"   and i :: "'a" shows "pmf (bind_pmf N f) i = measure_pmf.expectation N (\<lambda>x. pmf (f x) i)"
split_face_distinct2': fixes oldF :: "face"   and ram1 :: "nat"   and ram2 :: "nat"   and newVertexList :: "nat list" assumes "pre_split_face oldF ram1 ram2 newVertexList" shows "distinct (vertices (snd (split_face oldF ram1 ram2 newVertexList)))"
MI_pred_def: fixes K :: "('c \<Rightarrow> univ \<Rightarrow> bool) \<times> ('a \<Rightarrow> univ list \<Rightarrow> univ) \<times> ('b \<Rightarrow> univ list \<Rightarrow> bool)" shows "MI_pred K \<equiv> IInfModel (fst3 K) (snd3 K) (trd3 K)"
Quotient_uprod: shows "Quotient eq_upair abs_uprod rep_uprod cr_uprod"
assms: shows "R[x\<turnstile>n>y] = OrL v.M w.N u"
zero_under: fixes x :: "'b" assumes "x \<in> Field s" shows "s.zero \<in> s.under x"
simps_1: shows "f_bnd False = 2 ^ (m - 1) * N ^ m * m"
sorted_list_of_set_empty: shows "ord_term_lin.sorted_list_of_set {} = []"
independent_explicit: fixes B :: "'c set" shows "vs2.independent B = (finite B \<and> (\<forall>c. (\<Sum>v\<in>B. c v *b v) = (0::'c) \<longrightarrow> (\<forall>v\<in>B. c v = (0::'a))))"
permutation_insert_permutes: fixes q :: "nat \<Rightarrow> nat"   and n :: "nat"   and i :: "nat"   and j :: "nat" assumes "q permutes {0..<n}"   and "i < Suc n"   and "j < Suc n" shows "permutation_insert i j q permutes {0..<Suc n}"
canonical_fun_def: fixes U :: "'a set"   and x :: "'b" shows "canonical_fun U x = \<lfloor>U, x\<rfloor>"
a_closed: fixes H :: "'e set"   and G :: "('e, 'f) ring_scheme"   and x :: "'e"   and y :: "'e" assumes "additive_subgroup H G"   and "x \<in> H"   and "y \<in> H" shows "x \<oplus>\<^bsub>G\<^esub> y \<in> H"
Cond_not_s\<^sub>1: fixes s\<^sub>1 :: "stmt"   and e :: "expr"   and s\<^sub>2 :: "stmt" shows "s\<^sub>1 \<noteq> Cond e s\<^sub>1 s\<^sub>2"
sabove_halfspace_def: fixes sctn :: "'b sctn" shows "sabove_halfspace sctn = Collect (gt_halfspace sctn)"
alpha: shows "\<langle>\<alpha>\<^sub>L,P\<^sub>L'\<rangle> = \<langle>L_action.Act \<alpha>,EF (L (\<alpha>, F, f), P')\<rangle>"
B_trusts_NS3_lemma: fixes A :: "agent"   and B :: "agent"   and evs :: "event list"   and NB :: "nat"   and NA :: "nat" assumes "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> ns_public"   and "Crypt (pubK B) (Nonce NB) \<in> parts (knows Spy evs)"   and "Says B A (Crypt (pubK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs" shows "Says A B (Crypt (pubK B) (Nonce NB)) \<in> set evs"
real_of_rat_remdups_adj: fixes xs :: "rat list" shows "remdups_adj (map real_of_rat xs) = map real_of_rat (remdups_adj xs)"
gen_dfs_pre_imp_post_brk: fixes E :: "('a \<times> 'a) set"   and U :: "'a set"   and S :: "'a set"   and V :: "'a set"   and u0 :: "'a" assumes "gen_dfs_pre E U S V u0" shows "gen_dfs_post E U S V u0 (insert u0 V) True"
prim_invar2_init_def: fixes r :: "'a"   and Q :: "'a \<Rightarrow> enat"   and \<pi> :: "'a \<Rightarrow> 'a option" shows "Prim2.prim_invar2_init r Q \<pi> \<equiv> Q = Prim2.initQ r \<and> \<pi> = Prim2.init\<pi>"
not_essential_transform: fixes g :: "complex \<Rightarrow> complex"   and z :: "complex"   and f :: "complex \<Rightarrow> complex" assumes "not_essential g z"   and "\<forall>\<^sub>F w in at z. g w = f w" shows "not_essential f z"
elims: fixes x :: "'c regular \<times> 'c dense Filters.filter"   and y :: "'c regular \<times> 'c dense Filters.filter" assumes "stone_phi.pairs_uminus x = y" obtains xa :: "'c regular" and ya :: "'c dense Filters.filter" where "x = (xa, ya)" and "y = (- xa, stone_phi xa)"
succ_imp_graph_correct: fixes cnf :: "lit uprod set"   and l :: "lit" assumes "finite cnf" shows "succ_imp_graph cnf l = {l'. (l, l') \<in> imp_graph cnf}"
g'_in_carrier: fixes \<G> :: "'a cyclic_group"   and x :: "nat" assumes "chaum_ped_\<Sigma> \<G>" shows "chaum_ped_\<Sigma>_base.g' \<G> x \<in> carrier \<G>"
ras_is_empty_def: fixes s :: "'c array \<times> nat" shows "ras_is_empty s \<equiv> ras_length s = 0"
config'_rand_snoc: fixes A :: "('c list \<Rightarrow> 'd pmf) \<times> ('c list \<times> 'd \<Rightarrow> 'c \<Rightarrow> ((nat \<times> nat list) \<times> 'd) pmf)"   and s :: "('c list \<times> 'd) pmf"   and rs :: "'c list"   and r :: "'c" shows "config'_rand A s (rs @ [r]) = config'_rand A s rs \<bind> (\<lambda>s. snd A s r \<bind> (\<lambda>(a, is'). return_pmf (step (fst s) r a, is')))"
rawpsubstT_Var_in: fixes txs :: "('trm \<times> 'var) list"   and s :: "'trm"   and y :: "'var" assumes "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm"   and "distinct (map snd txs)"   and "(s, y) \<in> set txs"   and "\<And>i j. \<lbrakk>i < j; j < length txs\<rbrakk> \<Longrightarrow> snd (txs ! j) \<notin> FvarsT (fst (txs ! i))" shows "rawpsubstT (Var y) txs = s"
seq_char: fixes g :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr"   and f :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr" shows "A2_B.seq g f = (A2_B.arr f \<and> A2_B.arr g \<and> A2_B.Dom g = A2_B.Cod f)"
lm050: fixes a :: "('d \<times> 'c set) set" shows "Range (pseudoAllocation a) = finestpart (\<Union> (Range a))"
l_set_val_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_val :: "(_) character_data_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_val_locs :: "(_) character_data_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   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" shows "l_set_val_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_val set_val_locs known_ptr get_child_nodes get_child_nodes_locs \<equiv> l_set_val\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_val set_val_locs \<and> l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr get_child_nodes get_child_nodes_locs"
analyzed_closed_mod_timpls'_def: fixes M :: "(('fun, 'atom, 'sets) prot_fun, 'a) Term.term list"   and TI :: "('sets set \<times> 'sets set) list" shows "analyzed_closed_mod_timpls' M TI \<equiv> let f = list_all (intruder_synth_mod_timpls' M TI); g = \<lambda>t. if f (fst (Ana t)) then f (snd (Ana t)) else \<forall>s\<in>comp_timpl_closure {t} (set TI). case Ana s of (K, R) \<Rightarrow> f K \<longrightarrow> f R in list_all g M"
restrict_spmf_def: fixes p :: "'h spmf"   and A :: "'h set" shows "p \<upharpoonleft> A = map_pmf (\<lambda>x. x \<bind> (\<lambda>y. if y \<in> A then Some y else None)) p"
F_first_is_continuous: shows "continuous_on Dx_pair (\<lambda>a. F a \<bullet> j)"
drop_eq_Nil: fixes n :: "nat"   and xs :: "'b list" shows "(drop n xs = []) = (length xs \<le> n)"
Some: shows "lass_val_of e = \<lfloor>v\<rfloor>"
assms_3: fixes n :: "nat" shows "u n \<in> space\<^sub>N N"
is_mult_sum4sq_nat: fixes x :: "nat"   and y :: "nat" assumes "is_sum4sq_nat x"   and "is_sum4sq_nat y" shows "is_sum4sq_nat (x * y)"
wp2While: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "(if bval b s then wp\<^sub>2 c (wp\<^sub>2 (WHILE b DO c) Q) s else Q s) + 1 = wp\<^sub>2 (WHILE b DO c) Q s"
simps_2: fixes x :: "'a"   and xs :: "[:'a:]"   and ys :: "[:'a:]" assumes "x \<noteq> \<bottom>"   and "xs \<noteq> \<bottom>" shows "(x :# xs) :@ ys = x :# xs :@ ys"
leg1_composite_2: shows "\<mu>\<nu>_\<pi>.leg1 = \<mu>.leg1 \<cdot> Prj\<^sub>1\<^sub>1"
fully_pac_assn_def: shows "fully_pac_assn = list_assn (hr_comp (pac_step_rel_assn uint64_nat_assn poly_assn string_assn) (p2rel (\<langle>nat_rel, fully_unsorted_poly_rel O mset_poly_rel, var_rel\<rangle>pac_step_rel_raw)))"
rep_eq: fixes x :: "'c \<Longrightarrow> 'd"   and xa :: "('e, 'd) uexpr" shows "(x \<sharp> xa) = (\<forall>b v. \<lbrakk>xa\<rbrakk>\<^sub>e (put\<^bsub>x\<^esub> b v) = \<lbrakk>xa\<rbrakk>\<^sub>e b)"
subspace_add: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set"   and x :: "'b"   and y :: "'b" assumes "module scale"   and "module.subspace scale S"   and "x \<in> S"   and "y \<in> S" shows "x + y \<in> S"
subst_agreement: fixes t :: "('a, 'c) Term.term"   and r :: "'c \<Rightarrow> ('a, 'b) Term.term"   and s :: "'c \<Rightarrow> ('a, 'b) Term.term" shows "(t \<cdot> r = t \<cdot> s) = (\<forall>v\<in>fv t. Var v \<cdot> r = Var v \<cdot> s)"
IH_x_less: shows "(\<Sum>i<n. x_less div base ^ i mod base * base ^ i) = x_less"
theorem24xxiv_pp: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- (x \<sqinter> y) \<squnion> - - (x \<sqinter> z) = - (x \<sqinter> (y \<sqinter> - z)) \<squnion> - - (x \<sqinter> z)"
add0: fixes k :: "int"   and y :: "'b"   and t :: "'b tree23" shows "add0 k y t = gtree (add k y t)"
abs_eq: fixes xa :: "j \<Rightarrow> i \<Rightarrow> bool"   and x :: "j \<Rightarrow> i \<Rightarrow> bool" shows "make\<o> xa \<^bold>\<rightarrow> make\<o> x = make\<o> (\<lambda>s w. s = dj \<and> (xa dj w \<longrightarrow> x dj w) \<or> s \<noteq> dj \<and> I_IMPL s (xa s) (x s) w)"
assms_1: fixes u :: "'t" assumes "u \<prec>\<^sub>t v" shows "lookup p u = (0::'b)"
preserves_cod: fixes f :: "'a1" assumes "A1.arr f" shows "A2_B.cod (\<tau> f) = G (A1.cod f)"
filter_filter_mset_ss_member: fixes x :: "'a"   and y :: "'a"   and A :: "'a set multiset" shows "filter_mset ((\<subseteq>) {x, y}) A = filter_mset ((\<subseteq>) {x, y}) (filter_mset ((\<in>) x) A)"
ldistinct_lfilterI: fixes xs :: "'c llist"   and P :: "'c \<Rightarrow> bool" assumes "ldistinct xs" shows "ldistinct (lfilter P xs)"
one_aezfun_def: shows "1 \<equiv> (1::'a) \<delta>\<delta> (0::'b)"
lessThan_iff: fixes i :: "'e"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and k :: "'e" shows "(i \<in> ord.lessThan less k) = less i k"
dgrad_p_set_le_trans: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'd) set"   and G :: "('t \<Rightarrow>\<^sub>0 'd) set"   and H :: "('t \<Rightarrow>\<^sub>0 'd) set" assumes "dgrad_p_set_le d F G"   and "dgrad_p_set_le d G H" shows "dgrad_p_set_le d F H"
prj_simps_16: shows "C.cod gfg.Prj\<^sub>1 = g.apex"
a3: fixes i :: "nat" shows "g i \<in> f (Suc i) \<and> (\<exists>q. is_red {g i} q \<and> \<not> is_red (f i) q)"
G_def: shows "G = (\<lambda>x. A x + B x)"
paths_with_new_start_in_v0: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and P_new :: "'c list"   and xs :: "'c list" assumes "DisjointPathsPlusOne G v0 v1 paths P_new"   and "xs \<in> DisjointPathsPlusOne.paths_with_new paths P_new" shows "hd xs = v0"
dist_le_max_Busemann_functions: fixes xi :: "'b Gromov_completion"   and eta :: "'b Gromov_completion"   and x :: "'b"   and y :: "'b" assumes "xi \<noteq> eta" shows "dist x y \<le> 2 * real_of_ereal (extended_Gromov_product_at y xi eta) + max (Busemann_function_at xi x y) (Busemann_function_at eta x y) + 2 * deltaG TYPE('b)"
hyp2_rep_abs_examples_1: shows "hyp2_rep (hyp2_abs 0) = 0"
intAg: shows "integrable M (\<lambda>x. indicat_real A (T x) * g x)"
chamber_distance_map: fixes C :: "'a set"   and D :: "'a set" assumes "domain.chamber C"   and "domain.chamber D" shows "codomain.chamber_distance (f ` C) (f ` D) = domain.chamber_distance C D"
simps_1: fixes x :: "'b"   and uu :: "'b"   and uv :: "('b \<times> 'b) set" shows "rel_chain_fun 0 x uu uv = x"
simps_2: fixes C :: "'b \<Rightarrow> 'c \<Rightarrow> 'd option"   and a :: "'c" shows "applied_rule C a [] = \<bottom>"
bij_eq_iff: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'c" assumes "bij f" shows "(f x = f y) = (x = y)"
atMost_subset_iff: fixes x :: "'e"   and y :: "'e" shows "({..x} \<subseteq> {..y}) = (x \<le> y)"
subset: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'c set"   and B :: "'c set" assumes "semilattice_set f"   and "finite A"   and "B \<noteq> {}"   and "B \<subseteq> A" shows "f (semilattice_set.F f B) (semilattice_set.F f A) = semilattice_set.F f A"
image_add_atLeastAtMost': fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and k :: "'b"   and i :: "'b"   and j :: "'b" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "(\<lambda>n. plus n k) ` ord.atLeastAtMost less_eq i j = ord.atLeastAtMost less_eq (plus i k) (plus j k)"
rel_word_rel: fixes v :: "'d \<Rightarrow> ('c \<times> 'c) set"   and r :: "'d rexp" shows "rel v r = \<Union> (word_rel v ` lang r)"
thetaFDW0_RetrT: shows "thetaFDW0 \<subseteq> RetrT thetaFDW0"
pairwise_aux: fixes x :: "'a"   and xs :: "'a list"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "(\<forall>j<length (x # xs). \<forall>i<j. P ((x # xs) ! i) ((x # xs) ! j)) = ((\<forall>j<length xs. P x (xs ! j)) \<and> (\<forall>j<length xs. \<forall>i<j. P (xs ! i) (xs ! j)))"
pdg_dg_prod_Arr_in_Vset: fixes \<beta> :: "V" assumes "\<Z> \<beta>"   and "\<alpha> \<in>\<^sub>\<circ> \<beta>" shows "dg_prod I \<AA>\<lparr>Arr\<rparr> \<in>\<^sub>\<circ> Vset \<beta>"
oarrivemsg_simps_2: fixes P :: "(nat \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'd"   and d :: "nat"   and m :: "nat" shows "oarrivemsg P \<sigma> (d:deliver(m)) = True"
A_card_eq: shows "card A = nat ((int (nat p) - 1) div 2)"
sup_insertI: fixes x :: "'a"   and A :: "'a set"   and P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "\<And>s. is_lub L s (insert x A) \<Longrightarrow> P s"   and "is_lub L a A"   and "x \<in> carrier L"   and "A \<subseteq> carrier L" shows "P (\<Squnion>insert x A)"
simps_2: fixes p :: "('b \<times> 'a \<times> 'b) list"   and w :: "'a" shows "mpath_weight' (Some (p, w)) = Num w"
all_oid_diff: fixes \<tau> :: "'d state \<times> 'd state"   and x :: "'d state \<times> 'd state \<Rightarrow> 'e"   and X :: "'d state \<times> 'd state \<Rightarrow> Set('e)" assumes "\<tau> \<Turnstile> \<delta> x"   and "\<tau> \<Turnstile> \<delta> X"   and "\<And>x. x \<in> \<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (X \<tau>)\<rceil>\<rceil> \<Longrightarrow> x \<noteq> null" shows "(\<tau> \<Turnstile> X->forAll\<^sub>S\<^sub>e\<^sub>t(a|not (StrictRefEq\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t x a))) = (oid_of (x \<tau>) \<notin> oid_of ` \<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (X \<tau>)\<rceil>\<rceil>)"
substAbs_vsubstAbs_compose2: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set"   and Z :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and ys :: "'varSort"   and zs :: "'varSort"   and y :: "'var"   and z :: "'var"   and y1 :: "'var" assumes "goodAbs A"   and "good Z"   and "ys \<noteq> zs \<or> y \<noteq> z"   and "fresh ys y Z" shows "A $[y1 // y]_ys $[Z / z]_zs = A $[Z / z]_zs $[Var ys y1 #[Z / z]_zs / y]_ys"
FL_valid_Act: fixes P :: "'state"   and f :: "'effect"   and \<alpha> :: "'act"   and x :: "('a, 'pred, 'act, 'effect) formula" shows "P \<Turnstile> Act f \<alpha> x = (\<exists>\<alpha>' x' P'. Act f \<alpha> x = Act f \<alpha>' x' \<and> \<langle>f\<rangle>P \<rightarrow> \<langle>\<alpha>',P'\<rangle> \<and> P' \<Turnstile> x')"
assms: fixes d :: "'a \<Rightarrow> nat"   and gs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and m :: "nat"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and sps :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and data :: "nat \<times> 'd"   and p :: "('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c"   and q :: "('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c" assumes "dickson_grading d"   and "fst ` set gs \<subseteq> dgrad_p_set d m"   and "is_Groebner_basis (fst ` set gs)"   and "fst ` set bs \<subseteq> dgrad_p_set d m"   and "set ps \<subseteq> set bs \<times> (set gs \<union> set bs)"   and "sps \<noteq> []"   and "set sps \<subseteq> set ps"   and "unique_idx (gs @ bs) data"   and "(p, q) \<in> set sps"   and "fst p \<noteq> 0"   and "fst q \<noteq> 0" shows "crit_pair_cbelow_on d m (fst ` (set gs \<union> set bs) \<union> fst ` set (fst (compl gs bs (ps -- sps) sps data))) (fst p) (fst q)"
assm: shows "\<forall>X. \<Sqinter>{g x |x. x \<in> X} = g (\<Sqinter>X)"
sorted_append: fixes xs :: "'b list"   and ys :: "'b list" shows "sorted (xs @ ys) = (sorted xs \<and> sorted ys \<and> (\<forall>x\<in>set xs. \<forall>y\<in>set ys. x \<le> y))"
supteq_ctxtE: fixes s :: "('d, 'e) Term.term"   and t :: "('d, 'e) Term.term" assumes "s \<unrhd> t" obtains C :: "('d, 'e) ctxt" where "s = C\<langle>t\<rangle>"
lessThan_conv: fixes A :: "nat set"   and n :: "nat" assumes "card A = n"   and "\<forall>x\<in>A. x < n" shows "A = {..<n}"
MxS: shows "{y \<in> S. y \<le> x} \<subseteq> S"
AssertionStatEqTrans: fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and \<Psi> :: "'d"   and \<Psi>' :: "'d"   and \<Psi>'' :: "'d" assumes "assertionAux SCompose SImp SBottom SChanEq"   and "assertionAux.AssertionStatEq SImp \<Psi> \<Psi>'"   and "assertionAux.AssertionStatEq SImp \<Psi>' \<Psi>''" shows "assertionAux.AssertionStatEq SImp \<Psi> \<Psi>''"
bernpoly_of_real: fixes n :: "nat"   and x :: "real" shows "bernpoly n (of_real x) = of_real (bernpoly n x)"
cases: fixes x :: "'a simple_match" obtains uu_ :: "iface" and uv_ :: "iface" and uw_ :: "'a word \<times> nat" and ux_ :: "'a word \<times> nat" and uy_ :: "protocol" and sps1 :: "16 word" and sps2 :: "16 word" and dps1 :: "16 word" and dps2 :: "16 word" where "x = \<lparr>iiface = uu_, oiface = uv_, src = uw_, dst = ux_, proto = uy_, sports = (sps1, sps2), dports = (dps1, dps2)\<rparr>"
angle_action_def: fixes P :: "(nat \<Rightarrow> 'd) \<Rightarrow> bool"   and v :: "'d \<Rightarrow> 'e" shows "\<diamond>\<langle>P\<rangle>_v \<equiv> \<not> \<box>[\<not> P]_v"
simpnum_ci: fixes bs :: "real list"   and t :: "MIR.num" shows "Inum bs (simpnum t) = Inum bs t"
not_arr_null: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "partial_magma C" shows "\<not> partial_magma.arr C (partial_magma.null C)"
sign_compose: fixes p :: "'b \<Rightarrow> 'b"   and q :: "'b \<Rightarrow> 'b" assumes "permutation p"   and "permutation q" shows "sign (p \<circ> q) = sign p * sign q"
expected_value_is_utility_function: fixes outcomes :: "'b set"   and x :: "'b pmf"   and y :: "'b pmf"   and \<R> :: "('b pmf \<times> 'b pmf) set"   and u :: "'b \<Rightarrow> real" assumes "finite outcomes"   and "outcomes \<noteq> {}"   and "x \<in> \<P> outcomes"   and "y \<in> \<P> outcomes"   and "ordinal_utility (\<P> outcomes) \<R> (\<lambda>x. measure_pmf.expectation x u)" shows "(measure_pmf.expectation y u \<le> measure_pmf.expectation x u) = x \<succeq>[\<R>] y"
map_invar_impl_def: fixes m :: "nat \<times> nat \<times> nat \<Rightarrow> nat option"   and s :: "bdd" shows "mi.map_invar_impl m s = (\<forall>ii ti ei ri. m (ii, ti, ei) = Some ri \<longrightarrow> (\<exists>i t e. (ri, ifex_ite_opt i t e) \<in> Rmi s \<and> (ii, i) \<in> Rmi s \<and> (ti, t) \<in> Rmi s \<and> (ei, e) \<in> Rmi s))"
elims_3: fixes x :: "vname"   and xa :: "vname" assumes "\<not> x < xa" obtains n1 :: "nat" and n2 :: "nat" where "x = R n1__" and "xa = I n2__"   | n1 :: "nat" and n2 :: "nat" where "x = I n1a__" and "xa = I n2a__" and "\<not> n1a__ < n2a__"   | n1 :: "nat" and n2 :: "nat" where "x = R n1" and "xa = R n2" and "\<not> n1 < n2"
kbo_as_wpo: fixes n :: "nat"   and t :: "('f, 'c) Term.term"   and s :: "('f, 'c) Term.term" assumes "bounded_arity n (funas_term t)" shows "kbo s t = WPO.wpo.wpo n weight_S weight_NS (\<lambda>f g. (pr_strict' f g, pr_weak' f g)) (\<lambda>(f, n). n = 0 \<and> least f) full_status (\<lambda>_. Lex) False (\<lambda>f. False) s t"
sub: fixes a :: "'a" assumes "a \<in> H" shows "a \<in> G"
vdiff_dotprod_distr: fixes b :: "nat list"   and w :: "nat list"   and v :: "nat list" assumes "length b = length w"   and "v \<le>\<^sub>v w" shows "(w -\<^sub>v v) \<bullet> b = w \<bullet> b - v \<bullet> b"
dickson_leI: fixes ord :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ord_strict :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and d :: "'a \<Rightarrow> nat"   and s :: "'a"   and m :: "nat"   and t :: "'a" assumes "gd_powerprod ord ord_strict"   and "d s \<le> m"   and "d t \<le> m"   and "ord s t" shows "gd_powerprod.dickson_le ord d m s t"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'a"   and keymode :: "keymode" shows "P (case keymode of Signature \<Rightarrow> f1 | Encryption \<Rightarrow> f2) = ((keymode = Signature \<longrightarrow> P f1) \<and> (keymode = Encryption \<longrightarrow> P f2))"
basechambers_half_chamber_systems_1: shows "C0 \<in> f \<turnstile> folding_g.\<C>"
intro: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a' :: "'a" assumes "partial_magma.ide C a'" shows "yoneda_lemma_for_hom_axioms C a'"
open_union: fixes F :: "'a set set" assumes "\<And>x. x \<in> F \<Longrightarrow> is_open x" shows "is_open (\<Union>x\<in>F. x)"
first_step: fixes u :: "nat \<Rightarrow> 'a"   and c :: "'a" assumes "\<And>n. u n \<in> S"   and "u \<longlonglongrightarrow> c"   and "c \<in> T" shows "(\<lambda>n. f (u n)) \<longlonglongrightarrow> f c"
Image_Un: fixes R :: "('f \<times> 'e) set"   and A :: "'f set"   and B :: "'f set" shows "R `` (A \<union> B) = R `` A \<union> R `` B"
BoxDmd: fixes F :: "'a \<Rightarrow> bool"   and G :: "behavior \<Rightarrow> bool" shows "\<turnstile> \<box>F \<and> \<diamond>G \<longrightarrow> \<diamond>(\<box>F \<and> G)"
try_bind_spmf_lossless2: fixes q :: "'a spmf"   and p :: "'b spmf"   and f :: "'b \<Rightarrow> 'a spmf" assumes "lossless_spmf q" shows "TRY p \<bind> f ELSE q = TRY p \<bind> (\<lambda>x. TRY f x ELSE q) ELSE q"
Eq: shows "the_elem (h ` X) = the_elem (h ` Y)"
bigo_const1: fixes c :: "'b" shows "(\<lambda>x. c) \<in> O(\<lambda>x. 1::'b)"
assms_3: shows "list_all (typ_ok \<Theta>) (map snd insts)"
thin_Un: fixes H :: "fmla set"   and A :: "fmla"   and H' :: "fmla set" assumes "H \<turnstile> A" shows "H \<union> H' \<turnstile> A"
lens_unit_plus_sublens_1: fixes X :: "'d \<Longrightarrow> 'e" shows "X \<subseteq>\<^sub>L 0\<^sub>L +\<^sub>L X"
less_sat_def: fixes x :: "'a sat"   and y :: "'a sat" shows "(x < y) = (nat_of x < nat_of y)"