hcju/isabelleps huggingface dataset:
chain_subset_def: fixes C :: "'a set set" shows "chain\<^sub>\<subseteq> C = (\<forall>A\<in>C. \<forall>B\<in>C. A \<subseteq> B \<or> B \<subseteq> A)"
card_eq_SucD: fixes A :: "'a set"   and k :: "nat" assumes "card A = Suc k" shows "\<exists>b B. A = insert b B \<and> b \<notin> B \<and> card B = k \<and> (k = 0 \<longrightarrow> B = {})"
exec_iff_execn: fixes s :: "state"   and c :: "com"   and t :: "state" shows "(s -c\<rightarrow> t) = (\<exists>n. s -c-n\<rightarrow> t)"
grey_protects_white_def: fixes g :: "'h"   and w :: "'h" shows "g grey_protects_white w = (\<lambda>s. grey g s \<and> (g has_white_path_to w) s)"
same_span_imp_card_eq_dim_span: fixes V :: "'b vec set"   and n :: "nat"   and W :: "'b vec set" assumes "V \<subseteq> carrier_vec n"   and "W \<subseteq> carrier_vec n"   and "LinearCombinations.module.span class_ring (module_vec TYPE('b) n) V = LinearCombinations.module.span class_ring (module_vec TYPE('b) n) W"   and "\<not> module.lin_dep class_ring (module_vec TYPE('b) n) V" shows "card V = vec_space.dim_span n W"
comm_group_vec: fixes n :: "nat" shows "comm_group (monoid_vec TYPE('b) n)"
is_Plus_def: fixes rexp :: "'a rexp" shows "is_Plus rexp = (\<exists>x41 x42. rexp = Plus x41 x42)"
residual_abs_rep: fixes res :: "('b, 'c) residual" shows "abs_residual (rep_residual res) = res"
inv\<^sub>1_def: fixes P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and V :: "'a set" shows "inv\<^sub>1 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V = (bp (P\<^sub>1 \<union> wrap B\<^sub>1 \<union> P\<^sub>2 \<union> wrap B\<^sub>2 \<union> {{v} |v. v \<in> V}) \<and> \<Union> (P\<^sub>1 \<union> wrap B\<^sub>1 \<union> P\<^sub>2 \<union> wrap B\<^sub>2) = U - V \<and> B\<^sub>1 \<notin> P\<^sub>1 \<union> P\<^sub>2 \<union> wrap B\<^sub>2 \<and> B\<^sub>2 \<notin> P\<^sub>1 \<union> wrap B\<^sub>1 \<union> P\<^sub>2 \<and> (P\<^sub>1 \<union> wrap B\<^sub>1) \<inter> (P\<^sub>2 \<union> wrap B\<^sub>2) = {})"
flow0_inj_on: fixes t :: "real"   and t' :: "real"   and x :: "'a" assumes "t \<le> t'"   and "{t..t'} \<subseteq> existence_ivl0 x"   and "\<And>s. \<lbrakk>t < s; s \<le> t'\<rbrakk> \<Longrightarrow> flow0 x s \<noteq> flow0 x t" shows "inj_on (flow0 x) {t..t'}"
assms_3: shows "\<mu> \<star> src f = (f \<star> \<i>[src f]) \<cdot> \<a>[f, src f, src f]"
with_w_def: fixes quorum_of :: "'a \<Rightarrow> 'a set \<Rightarrow> bool" shows "with_w quorum_of \<equiv> personal_quorums quorum_of"
inv\<^sub>3E_1: fixes P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and S :: "'a set"   and L :: "'a set" assumes "inv\<^sub>3 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 S L" shows "inv\<^sub>2 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 S L"
aux: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c \<Rightarrow> bool"   and B :: "'c \<Rightarrow> bool" shows "{f n |n. A n \<or> B n} = {f n |n. A n} \<union> {f n |n. B n}"
map_ide_simp: fixes a :: "'a" assumes "A.ide a" shows "local.map a = B.inv (\<tau> a)"
in_quotientP_imp_subset: fixes P :: "'c \<Rightarrow> bool"   and \<phi> :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and X :: "'c set" assumes "equivP P \<phi>"   and "(P /// \<phi>) X" shows "X \<subseteq> Collect P"
zero_rel_def: fixes S :: "'c set"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c" assumes "Comm_Ring.quotient_ring S R addition multiplication zero unit" shows "quotient_ring.zero_rel S R addition multiplication zero unit = quotient_ring.frac S R addition multiplication zero zero unit"
unit_part: shows "\<guillemotleft>f \<star> \<iota> : f \<star> a \<star> a \<Rightarrow>\<^sub>S f \<star> a\<guillemotright> \<and> S.iso (f \<star> \<iota>)"
components_subset_struct: fixes sel :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ap :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ab :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and compl :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and sps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data' :: "nat \<times> 'g" assumes "struct_spec sel ap ab compl"   and "ps \<noteq> []"   and "sps = sel gs bs ps data"   and "hs = fst (add_indices (compl gs bs (ps -- sps) sps data) data)" shows "component_of_term ` Keys (args_to_set (gs, ab gs bs hs data', ap gs bs (ps -- sps) hs data')) \<subseteq> component_of_term ` Keys (args_to_set (gs, bs, ps))"
lasso_accepted: fixes m :: "'a"   and L :: "'Q lasso" assumes "gb_graph.is_lasso (to_gbg_ext m) L" shows "gb_graph.is_acc_run (to_gbg_ext m) (run_of_lasso L)"
perfect_hiding: fixes \<eta> :: "nat"   and \<A> :: "('grp \<times> nat \<times> nat \<Rightarrow> ((nat \<times> nat) \<times> 'b) spmf) \<times> ('grp \<Rightarrow> 'b \<Rightarrow> bool spmf)" shows "okamoto_\<Sigma>_commit.abstract_com.perfect_hiding_ind_cpa \<eta> \<A>"
nths_shift: fixes S :: "nat set"   and xs :: "'b list"   and ys :: "'b list" assumes "\<forall>i\<in>S. length xs \<le> i" shows "nths (xs @ ys) S = nths ys {x - length xs |x. x \<in> S}"
wf_NFAND: fixes \<Phi> :: "('a, 'k) aformula set"   and idx :: "'n" assumes "finite \<Phi>" shows "local.wf idx (NFAND \<Phi>) = (\<forall>\<phi>\<in>\<Phi>. local.wf idx \<phi>)"
cascadingaxiom: fixes s :: "'b"   and a :: "'a"   and b :: "'a" shows "s \<circ> (a ; b) = s \<circ> a ; \<lambda> (s, a) \<circ> b"
assms: shows "\<pi>: (WHILE b DO c, s) \<Rightarrow> t"
assms_2: fixes u :: "nat \<Rightarrow> 'a"   and b :: "'a" assumes "\<forall>n. u n \<in> S"   and "b \<in> T"   and "u \<longlonglongrightarrow> b" shows "(\<lambda>n. f (u n)) \<longlonglongrightarrow> f b"
Deriv_compl: fixes a :: "'b"   and A :: "'b list set" shows "Deriv a (- A) = - Deriv a A"
inject_2: fixes x21 :: "('a, 'b) com"   and x22 :: "('a, 'b) com"   and y21 :: "('a, 'b) com"   and y22 :: "('a, 'b) com" shows "(x21 ;; x22 = y21 ;; y22) = (x21 = y21 \<and> x22 = y22)"
aa2fs: shows "wfactors G (drop (Suc i) as') aa_2"
is_non_volatile_Write\<^sub>s\<^sub>b_def: fixes r :: "'b memref" shows "is_non_volatile_Write\<^sub>s\<^sub>b r = (case r of Write\<^sub>s\<^sub>b volatile a x v xa xb xc xd \<Rightarrow> \<not> volatile | _ \<Rightarrow> False)"
rec: fixes f1 :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and a :: "'e"   and b :: "'f" shows "old.rec_prod f1 (a, b) = f1 a b"
de_morgan_orthogonal_complement_inter: fixes A :: "'b set"   and B :: "'b set" assumes "closed_csubspace A"   and "closed_csubspace B" shows "orthogonal_complement (A \<inter> B) = orthogonal_complement A +\<^sub>M orthogonal_complement B"
add_mset_lt_lt_lt: fixes a :: "'a"   and b :: "'a"   and A :: "'a multiset"   and B :: "'a multiset" assumes "a < b"   and "A < B" shows "add_mset a A < add_mset b B"
isomorphic_symmetric: fixes f :: "'d"   and g :: "'d" assumes "D.isomorphic f g" shows "D.isomorphic g f"
Im_Arctan_of_real: fixes x :: "real" shows "Im (Arctan (complex_of_real x)) = 0"
integrableD_1: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> real" assumes "integrable M f" shows "f \<in> borel_measurable M"
simps_2: fixes n :: "enat"   and xs :: "'c llist" shows "(\<not> lnull (ltake n xs)) = (n \<noteq> 0 \<and> \<not> lnull xs)"
d_subset_N: fixes k :: "nat" shows "d k \<subseteq> N"
eex_mono: fixes F :: "(state \<Rightarrow> 'a) \<Rightarrow> behavior \<Rightarrow> bool"   and sigma :: "behavior"   and G :: "(state \<Rightarrow> 'a) \<Rightarrow> behavior \<Rightarrow> bool" assumes "(\<exists>\<exists> x. F x) sigma"   and "\<And>x. sigma \<Turnstile> F x \<longrightarrow> G x" shows "(\<exists>\<exists> x. G x) sigma"
12: shows "w\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w = bot"
ToCT_def: fixes f :: "'a \<times> 'a \<times> 'a" shows "T.ToCT f \<equiv> if CCC.arr f then local.Prod (fst f, local.Prod (fst (snd f), snd (snd f))) else null"
set_disjoint_single: fixes x :: "'a"   and B :: "'a set" shows "({x} \<inter> B = {}) = (x \<notin> B)"
MinRVars_def: fixes eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and min_rvar_incdec_eq :: "('a, 'b) Direction \<Rightarrow> ('a, 'b) state \<Rightarrow> nat \<times> linear_poly \<Rightarrow> 'a list + nat" shows "MinRVars eq_idx_for_lvar min_rvar_incdec_eq \<equiv> EqForLVar eq_idx_for_lvar \<and> MinRVarsEq min_rvar_incdec_eq"
lm083: fixes x :: "'c"   and X :: "'c set"   and g :: "('c \<times> 'd) set"   and Y :: "'d set" assumes "x \<notin> X"   and "g \<in> injections ({x} \<union> X) Y" shows "Outside' {x} g \<in> injections X Y"
zz1_def: shows "zz1 = restrict (I \<union> J) z1"
generate_ringE_2: fixes H :: "'a set"   and E :: "'a set" assumes "H \<subseteq> carrier R"   and "E = generate_ring R H" shows "H \<subseteq> E"
permutation_insert_expand: fixes i :: "'a"   and j :: "nat"   and p :: "'a \<Rightarrow> nat" shows "permutation_insert i j p \<equiv> \<lambda>i'. if i' < i then if p i' < j then p i' else Suc (p i') else if i' = i then j else if p (i' - (1::'a)) < j then p (i' - (1::'a)) else Suc (p (i' - (1::'a)))"
comp_left_conjugate: fixes x :: "'a" shows "conjugate ((*) x) ((*) (x\<^sup>T))"
summable_imp_Bseq: fixes f :: "nat \<Rightarrow> 'b" assumes "summable f" shows "Bseq f"
fst_of_sp_is_vert: fixes g :: "('c, 'd) rgraph_scheme"   and v1 :: "'c"   and es :: "'c edge list"   and v2 :: "'c" assumes "Graph.subpath g v1 es v2" shows "v1 \<in> Graph.vertices g"
f_last_message_hold_append_mult: fixes xs :: "'b message_af list"   and m :: "nat"   and k :: "nat"   and ys :: "'b message_af list" assumes "length xs = m * k" shows "(xs @ ys) \<longmapsto> k = xs \<longmapsto> k @ ys \<longmapsto> k"
restrict_indets_power: fixes p :: "('c option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and n :: "nat" shows "restrict_indets (p ^ n) = restrict_indets p ^ n"
nsg_l_rcs_eq: fixes N :: "'a set"   and a :: "'a" assumes "G \<triangleright> N"   and "a \<in> carrier G" shows "a \<diamondsuit> N = N \<bullet> a"
bind_def: fixes f :: "('e, 'f, 'h) prog"   and g :: "'h \<Rightarrow> ('e, 'f, 'g) prog" shows "f \<bind> g = Prog (\<lambda>h. case the_prog f h of Inl exception \<Rightarrow> Inl exception | Inr (x, h') \<Rightarrow> the_prog (g x) h')"
neg_le_neg_iff: fixes A :: "'a"   and B :: "'a" shows "(Neg A \<le> Neg B) = (A \<le> B)"
can_insert_def: shows "can_insert \<equiv> \<lambda>(l, r, buf). l < r"
lemma_2_prob_dom: fixes PROB :: "(('a, bool) fmap \<times> ('a, bool) fmap) set"   and s :: "('a, bool) fmap"   and as :: "(('a, bool) fmap \<times> ('a, bool) fmap) list" assumes "finite PROB"   and "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB"   and "2 ^ card (prob_dom PROB) - 1 < length as" shows "\<exists>as1 as2 as3. as1 @ as2 @ as3 = as \<and> exec_plan s (as1 @ as2) = exec_plan s as1 \<and> as2 \<noteq> []"
is_Const_def: fixes trm :: "trm" shows "is_Const trm = (\<exists>x2. trm = Const x2)"
PO_l2_inv5: shows "reach l2 \<subseteq> l2_inv5"
cases: fixes a :: "('b, 'c, 'd) exp" assumes "is_addr a" obtains aa :: "'d" where "a = addr aa"
cet_less_real: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e1 :: "cexpr"   and e2 :: "cexpr" assumes "\<Gamma> \<turnstile>\<^sub>c e1 : REAL"   and "\<Gamma> \<turnstile>\<^sub>c e2 : REAL" shows "\<Gamma> \<turnstile>\<^sub>c e1 <\<^sub>c e2 : BOOL"
Gcd_fin_eq_Gcd: fixes A :: "'a set" assumes "finite A" shows "Gcd\<^sub>f\<^sub>i\<^sub>n A = Gcd A"
V_nn_closed: fixes u :: "'c \<Rightarrow> 'a"   and t :: "'a" assumes "u \<in> V"   and "(0::'a) \<le> t" shows "u \<oplus> t \<in> V"
spec_ops_rem_cons: fixes x :: "'b \<times> 'c"   and xs :: "('b \<times> 'c) list"   and deps :: "'c \<Rightarrow> 'b set" assumes "spec_ops (x # xs) deps" shows "spec_ops xs deps"
intersects_sctns_def: fixes X' :: "'c set"   and sctns :: "'c sctn set" shows "intersects_sctns X' sctns = ASSUME (finite sctns) \<bind> (\<lambda>_. FOREACH\<^bsup>\<lambda>sctns' b. \<not> b \<longrightarrow> X' \<inter> \<Union> (plane_of ` (sctns - sctns')) = {}\<^esup> sctns (\<lambda>sctn b. op_intersects X' sctn \<bind> (\<lambda>b'. RETURN (b \<or> b'))) False)"
Gramian_determinant_def: fixes G :: "'a vec list"   and k :: "nat" shows "gs.Gramian_determinant G k = det (gs.Gramian_matrix G k)"
basic_subtype_OclAny_x: fixes \<sigma> :: "'a basic_type"   and P :: "bool" assumes "OclAny \<sqsubset>\<^sub>B \<sigma>" shows "P"
add_nonpos_neg: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq a zero"   and "less b zero" shows "less (plus a b) zero"
iTILL_div: fixes k :: "nat"   and n :: "nat" assumes "0 < k" shows "[\<dots>n] \<oslash> k = [\<dots>n div k]"
r_result_total: fixes t :: "nat"   and k :: "nat"   and x :: "nat" shows "eval r_result [t, k, x] \<down>"
anonymous_profile_update: fixes i :: "'agent"   and Ri' :: "'alt \<Rightarrow> 'alt \<Rightarrow> bool" assumes "i \<in> agents"   and "finite agents"   and "total_preorder_on alts Ri'" shows "anonymous_profile (R1(i := Ri')) = anonymous_profile R1 - {#weak_ranking (R1 i)#} + {#weak_ranking Ri'#}"
DINING_def: shows "DINING = (fold (\<lambda>i P. P|||FORK i) [1..<N] (FORK 0)||fold (\<lambda>i P. P|||RPHIL i) [1..<N] LPHIL0)"
num_reachable_def: fixes G :: "'b graph"   and v :: "'b" shows "FiniteGraph.num_reachable G v = card (FiniteGraph.succ_tran G v)"
true_inf_2: fixes P :: "'a upred" shows "P \<squnion> true = P"
constrains_UNIV2: fixes F :: "'b program"   and A :: "'b set" shows "F \<in> A co UNIV"
dropcptn_is_cptn: fixes j :: "nat"   and c :: "('b com option \<times> 'b) list" assumes "j < length c"   and "c \<in> cptn" shows "drop j c \<in> cptn"
gb_schema_incr_dgrad_p_set_isGB: fixes sel :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ap :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ab :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and compl :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g"   and upd :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> 'g \<Rightarrow> 'g"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list"   and data :: "'g" assumes "struct_spec sel ap ab compl"   and "compl_conn compl" shows "is_Groebner_basis (fst ` set (gb_schema_incr sel ap ab compl upd bs data))"
closed_sum_comm: fixes A :: "'a set"   and B :: "'a set" shows "A +\<^sub>M B = B +\<^sub>M A"
lhd_llist_of_tllist: fixes xs :: "('f, 'g) tllist" assumes "\<not> is_TNil xs" shows "lhd (llist_of_tllist xs) = thd xs"
map_add_restrict: fixes f :: "'c \<Rightarrow> 'd option"   and g :: "'c \<Rightarrow> 'd option" shows "f ++ g = f |` (- dom g) ++ g"
add_divide_eq_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and z :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "z \<noteq> zero" shows "plus x (divide y z) = divide (plus (times x z) y) z"
permutes_funpow: fixes p :: "'b \<Rightarrow> 'b"   and S :: "'b set"   and n :: "nat" assumes "p permutes S" shows "p ^^ n permutes S"
cone_Union: fixes f :: "'a set set" shows "(\<forall>s\<in>f. cone s) \<longrightarrow> cone (\<Union> f)"
minus_bit0_def: fixes x :: "'a bit0"   and y :: "'a bit0" shows "x - y = Abs_bit0' (Rep_bit0 x - Rep_bit0 y)"
qAbs_alphaAbs_iff: fixes xs :: "'e"   and x :: "'f"   and X :: "('c, 'd, 'e, 'f, 'g) qTerm"   and A' :: "('c, 'd, 'e, 'f, 'g) qAbs" shows "(qAbs xs x X $= A') = (\<exists>x' y X'. A' = qAbs xs x' X' \<and> y \<notin> {x, x'} \<and> qAFresh xs y X \<and> qAFresh xs y X' \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
rel_cases: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "('f, 'd) envT"   and b :: "('f, 'e) envT" assumes "envT.rel_envT R a b" obtains xa :: "'f \<Rightarrow> 'd" and ya :: "'f \<Rightarrow> 'e" where "a = EnvT xa" and "b = EnvT ya" and "((=) ===> R) xa ya"
closed_subset: fixes S :: "'b set"   and T :: "'b set" assumes "S \<subseteq> T"   and "closed S" shows "closedin (top_of_set T) S"
lrth: fixes xsa :: "'b list"   and xs :: "'c list"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and na :: "nat"   and n :: "nat" assumes "length xsa = length xs" shows "rotater1 (map2 f (rotater na xsa) (rotater n xs)) = map2 f (rotater1 (rotater na xsa)) (rotater1 (rotater n xs))"
prime_factorization_prod_mset: fixes A :: "'a multiset" assumes "(0::'a) \<notin># A" shows "prime_factorization (\<Prod>\<^sub># A) = \<Sum>\<^sub># (image_mset prime_factorization A)"
normal_subgroup_axioms: shows "normal_subgroup Ker G (\<cdot>) \<one>"
normal_ordering: fixes A :: "ordinal set" assumes "normal_set A" shows "normal (OrdinalVeblen.ordering A)"
mono_strong: fixes ys :: "'a list"   and xs :: "'a list"   and gt :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and gt' :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<forall>y\<in>set ys. \<forall>x\<in>set xs. gt y x \<longrightarrow> gt' y x"   and "ext gt ys xs" shows "ext gt' ys xs"
subspace_eq_real: shows "module_on.subspace (*\<^sub>R) = subspace"
inj_onI: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; f x = f y\<rbrakk> \<Longrightarrow> x = y" shows "inj_on f A"
group_hom_axioms: shows "group_hom (relative_homology_group (int n) (nsphere n) (equator n)) (relative_homology_group (int n) (nsphere n) (upper n)) (hom_induced (int n) (nsphere n) (equator n) (nsphere n) (upper n) id)"
callee_invariant': fixes callee :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'a) spmf"   and I :: "'a \<Rightarrow> bool"   and \<I> :: "('b, 'c) \<I>"   and y :: "'c"   and s' :: "'a"   and s :: "'a"   and x :: "'b" assumes "callee_invariant_on callee I \<I>"   and "(y, s') \<in> set_spmf (callee s x)"   and "I s"   and "x \<in> outs_\<I> \<I>" shows "I s' \<and> y \<in> responses_\<I> \<I> x"
an: shows "H \<turnstile> SyntaxN.All i (Neg (Var i IN t))"
intpow_inverse: fixes a :: "'c"   and n :: "int" shows "inverse a ^\<^sub>Z n = inverse (a ^\<^sub>Z n)"
Rep_lifted_pair: fixes x :: "('c, 'd) lifted_pair" shows "Rep_lifted_pair x \<in> {pf. \<forall>f. pf f \<in> triple.pairs (Rep_phi f)}"
state_rank_distinct: fixes p :: "'b"   and q :: "'b"   and n :: "nat"   and i :: "nat"   and j :: "nat" assumes "p \<noteq> q"   and "state_rank p n = Some i"   and "state_rank q n = Some j" shows "i \<noteq> j"
simps_2: fixes a :: "'b" shows "has_primitive (Match a) = True"
isShortestPath_def: fixes u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat" shows "isShortestPath u p v \<equiv> isPath u p v \<and> (\<forall>p'. isPath u p' v \<longrightarrow> length p \<le> length p')"
RangeI: fixes a :: "'d"   and b :: "'e"   and r :: "('d \<times> 'e) set" assumes "(a, b) \<in> r" shows "b \<in> Range r"
n: shows "\<forall>x\<in>fv (formula.Agg y \<omega> b f \<phi>). x < n"
chk_u_s_if: assumes "chksubs (>\<^sub>t) t s" shows "chksubs (>\<^sub>t) u s"
eq3: shows "\<phi> (\<omega> [^]\<^bsub>R\<^esub> ((M + 1) div 2)) = \<ominus>\<^bsub>S\<^esub> \<one>\<^bsub>S\<^esub>"
r_one: fixes x :: "'a vec" assumes "x \<in> carrier_vec n" shows "x + 0\<^sub>v n = x"
communicatingD1: fixes C :: "'b set"   and a :: "'b"   and b :: "'b" assumes "C \<in> UNIV // K2.communicating"   and "(a, b) \<in> K2.communicating"   and "a \<in> C" shows "b \<in> C"
compat_corr: shows "\<forall>I I'. oneone I \<longrightarrow> oneone I' \<longrightarrow> PlaneGraphIso.compat I' I = (let m = map_of I; m' = map_of I' in m \<subseteq>\<^sub>m m ++ m' \<and> inj_on (m ++ m') (dom (m ++ m')))"
assms: shows "\<Gamma> \<turnstile>\<^sub>v Sabs cs $\<^sub>s Svar n \<down> v"
strict_trans2: fixes b :: "'a list"   and a :: "'a list"   and c :: "'a list" assumes "strict_border b a"   and "border c b" shows "strict_border c a"
UnifierD_1: fixes f :: "'c"   and X :: "('c, 'd) Term.term list"   and \<sigma> :: "'d \<Rightarrow> ('c, 'd) Term.term"   and g :: "'c"   and Y :: "('c, 'd) Term.term list" assumes "Unifier \<sigma> (Fun f X) (Fun g Y)" shows "f = g"