hcju/isabelleps huggingface dataset:
len_lexext_rev_compat_list: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b"   and xs :: "'b list"   and xs' :: "'b list" assumes "gt y x" shows "len_lexext_rev gt (xs @ y # xs') (xs @ x # xs')"
dirichlet_prod_fds_nth_fds_right: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" shows "f \<star> ($) (fds g) = f \<star> g"
sq_mtx_times_vec_assoc: fixes A :: "'b sq_mtx"   and B :: "'b sq_mtx"   and x :: "(real, 'b) vec" shows "(A * B) *\<^sub>V x = A *\<^sub>V (B *\<^sub>V x)"
dvd_elements_mult_matrix_left: fixes A :: "'a mat"   and m :: "nat"   and n :: "nat"   and P :: "'a mat"   and x :: "'a" assumes "A \<in> carrier_mat m n"   and "P \<in> carrier_mat m m"   and "\<forall>i j. i < m \<and> j < n \<longrightarrow> x dvd A $$ (i, j)" shows "\<forall>i j. i < m \<and> j < n \<longrightarrow> x dvd (P * A) $$ (i, j)"
lselect_lselect'': fixes s :: "nat set"   and w :: "'b llist"   and t :: "nat set" assumes "\<And>i. i \<in> s \<Longrightarrow> enat i < llength w"   and "\<And>i. i \<in> t \<Longrightarrow> enat i < llength (lselect s w)" shows "lselect t (lselect s w) = lselect (nth_least s ` t) w"
B: fixes X :: "'a \<Rightarrow> 'b" shows "mono (P X)"
V4_def: shows "V4 = filter (\<lambda>v. vertextype g v \<noteq> (5, 0, 1)) V2"
notstutstep_nexnat1: fixes s :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "\<not> stutstep s n" shows "nextnat (s |\<^sub>s n) = 1"
card_non_empty_geq_one: fixes i :: "nat_int" shows "(i \<noteq> bot) = (1 \<le> |i|)"
invertible_qmpt_def: shows "invertible_qmpt \<equiv> bij T \<and> inv T \<in> M \<rightarrow>\<^sub>M M"
fail_def: fixes S :: "'e \<Rightarrow> 'd" shows "fail S = - S \<top>"
assms: shows "a \<in>\<^sub>\<circ> \<D>\<^sub>\<bullet> (r \<restriction>\<^sup>r\<^sub>\<bullet> A)"
MM1: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "\<turnstile> F = G" shows "\<turnstile> \<box>F = \<box>G"
is_ring_morphism: fixes U :: "'a set"   and V :: "'a set" assumes "is_open U"   and "is_open V"   and "V \<subseteq> U" shows "ring_homomorphism (\<rho> U V) (\<FF> U) +\<^bsub>U\<^esub> \<cdot>\<^bsub>U\<^esub> \<zero>\<^bsub>U\<^esub> \<one>\<^bsub>U\<^esub> (\<FF> V) +\<^bsub>V\<^esub> \<cdot>\<^bsub>V\<^esub> \<zero>\<^bsub>V\<^esub> \<one>\<^bsub>V\<^esub>"
proj2_rep_abs2: fixes v :: "(real, 3) vec" assumes "v \<noteq> 0" shows "\<exists>k. k \<noteq> 0 \<and> proj2_rep (proj2_abs v) = k *\<^sub>R v"
Collect_member: fixes dxs :: "'a set_dlist" shows "Collect (member dxs) = set (list_of_dlist dxs)"
mon_c_single: fixes fg :: "('g, 'h, 'i, 'f, 'j) flowgraph_rec_scheme"   and s :: "'g list" shows "mon_c fg {#s#} = mon_s fg s"
Collect_empty_eq: fixes P :: "'h \<Rightarrow> bool" shows "(Collect P = {}) = (\<forall>x. \<not> P x)"
eq3: assumes "\<not> r1 x y" shows "map f (merge_wrt r1 (x # xs) ys) = merge_wrt r2 (map f (x # xs)) (map f ys)"
add_block_alt: fixes b :: "'a set" shows "derived_incidence_sys.add_block b = add_mset b \<B>\<^sup>D"
poly_of_focus_zero: fixes x :: "'a" shows "poly_of_focus x 0 = 0"
int_pow_closed: fixes x :: "'a"   and i :: "int" assumes "x \<in> carrier G" shows "x [^] i \<in> carrier G"
IH: fixes xs :: "('a \<times> 'a option) list"   and ys :: "('a \<times> 'a option) list" assumes "insert_ops ops"   and "insert_seq start xs"   and "insert_ops xs"   and "insert_seq start ys"   and "insert_ops ys"   and "set xs \<subseteq> set ops"   and "set ys \<subseteq> set ops"   and "distinct (map fst xs @ map fst ys)"   and "fst (hd xs) < fst (hd ys)"   and "\<And>r. start = Some r \<Longrightarrow> r \<in> set (interp_ins ops)" shows "(\<forall>x\<in>set (map fst xs). \<forall>y\<in>set (map fst ys). list_order ops y x) \<and> (\<forall>r. start = Some r \<longrightarrow> (\<forall>a\<in>set (map fst xs). list_order ops r a) \<and> (\<forall>a\<in>set (map fst ys). list_order ops r a))"
enumerate_simps_1: fixes n :: "nat" shows "List.enumerate n [] = []"
inject_15: fixes x15 :: "expr"   and y15 :: "expr" shows "(throw x15 = throw y15) = (x15 = y15)"
distinct_adj_mapI: fixes xs :: "'c list"   and f :: "'c \<Rightarrow> 'd" assumes "distinct_adj xs"   and "inj_on f (set xs)" shows "distinct_adj (map f xs)"
pol_coeff_split: fixes f :: "nat \<times> (nat \<Rightarrow> 'a)" shows "pol_coeff S f = pol_coeff S (fst f, snd f)"
prems_1: shows "A $ i $ mod_type_class.from_nat k \<noteq> (0::'a)"
m_comm: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp" shows "x \<oplus> y = y \<oplus> x"
dghm_0_def: fixes \<AA> :: "V" shows "dghm_0 \<AA> = [[]\<^sub>\<circ>, []\<^sub>\<circ>, dg_0, \<AA>]\<^sub>\<circ>"
hyps: fixes xs :: "'a list" shows "length (fold remove1 ys xs) \<le> length xs"
int_pow_def2: fixes G :: "('a, 'b) monoid_scheme"   and a :: "'a"   and z :: "int" shows "a [^]\<^bsub>G\<^esub> z = (if z < 0 then inv\<^bsub>G\<^esub> (a [^]\<^bsub>G\<^esub> nat (- z)) else a [^]\<^bsub>G\<^esub> nat z)"
diff_numeral_special_12: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b" assumes "class.neg_numeral minus plus zero uminus" shows "minus (uminus one) (uminus one) = zero"
timpls_transformable_to'_inv_1: fixes TI :: "('e set \<times> 'e set) list"   and f :: "('f, 'g, 'e) prot_fun"   and T :: "(('f, 'g, 'e) prot_fun, 'h) Term.term list"   and g :: "('f, 'g, 'e) prot_fun"   and S :: "(('f, 'g, 'e) prot_fun, 'h) Term.term list" assumes "timpls_transformable_to' TI (Fun f T) (Fun g S)" shows "length T = length S"
vector_conv_compl: fixes v :: "'a" assumes "v \<sqinter> top = v" shows "top \<sqinter> - id v = - id v"
ins_primrec: fixes k :: "'c"   and p :: "real"   and t :: "('c \<times> real) tree" shows "ins k p t = rec_tree \<langle>\<langle>\<rangle>, (k, p), \<langle>\<rangle>\<rangle> (\<lambda>l z r l' r'. case z of (k1, p1) \<Rightarrow> if k < k1 then case l' of \<langle>\<rangle> \<Rightarrow> \<langle>\<rangle> | \<langle>l2, (k2, p2), r2\<rangle> \<Rightarrow> if 0 \<le> p2 - p1 then \<langle>\<langle>l2, (k2, p2), r2\<rangle>, (k1, p1), r\<rangle> else \<langle>l2, (k2, p2), \<langle>r2, (k1, p1), r\<rangle>\<rangle> else if k1 < k then case r' of \<langle>\<rangle> \<Rightarrow> \<langle>\<rangle> | \<langle>l2, (k2, p2), r2\<rangle> \<Rightarrow> if 0 \<le> p2 - p1 then \<langle>l, (k1, p1), \<langle>l2, (k2, p2), r2\<rangle>\<rangle> else \<langle>\<langle>l, (k1, p1), l2\<rangle>, (k2, p2), r2\<rangle> else \<langle>l, (k1, p1), r\<rangle>) t"
spd': shows "StrongPostdomination src trg knd (lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit) NewEntry NewExit"
split: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'c"   and f2 :: "'c"   and poscheckvariant :: "poscheckvariant" shows "P (case poscheckvariant of assign \<Rightarrow> f1 | check \<Rightarrow> f2) = ((poscheckvariant = assign \<longrightarrow> P f1) \<and> (poscheckvariant = check \<longrightarrow> P f2))"
rel_conversep: fixes R1 :: "'g \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'h \<Rightarrow> 'e \<Rightarrow> bool"   and R3 :: "'i \<Rightarrow> 'f \<Rightarrow> bool" shows "F2rel R1\<inverse>\<inverse> R2\<inverse>\<inverse> R3\<inverse>\<inverse> = (F2rel R1 R2 R3)\<inverse>\<inverse>"
simps_2: fixes Us :: "'a list" shows "combine [] Us = \<zero>"
rep_eq: fixes x :: "'a \<Rightarrow> real" shows "to_vec (sq_mtx_diag x) = diag_mat x"
A: fixes x :: "'a"   and \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "x \<in> A \<Gamma>" shows "x \<in> SAT \<Gamma> f"
hd_no_head: fixes u :: "'a list"   and b :: "'a list" assumes "u \<in> \<langle>\<C>\<rangle>"   and "hd (Dec \<C> u) \<noteq> b" shows "u \<in> no_head \<C> b"
carrier_mult_of: fixes R :: "('c, 'd) ring_scheme" shows "carrier (mult_of R) = carrier R - {\<zero>\<^bsub>R\<^esub>}"
RSpan_closed: fixes ms :: "'m list" assumes "set ms \<subseteq> M" shows "RSpan ms \<subseteq> M"
tmap_eq_tmap_conv_tllist_all2: fixes f :: "'c \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'b"   and xs :: "('c, 'd) tllist"   and f' :: "'e \<Rightarrow> 'a"   and g' :: "'f \<Rightarrow> 'b"   and ys :: "('e, 'f) tllist" shows "(tmap f g xs = tmap f' g' ys) = tllist_all2 (\<lambda>x y. f x = f' y) (\<lambda>x y. g x = g' y) xs ys"
simps_1: fixes S :: "nat \<Rightarrow> bool option" shows "share_all_until_volatile_write [] S = S"
rep_eq: fixes x :: "('d, 'c) sq_matrix"   and xa :: "('d, 'c) sq_matrix" shows "to_fun (x + xa) = (\<lambda>i j. to_fun x i j + to_fun xa i j)"
unit_simps_5: fixes a :: "'a" assumes "obj a" shows "cod \<i>[a] = a"
set_iterator_linord_def: fixes it :: "('d \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and S0 :: "'a set" shows "set_iterator_linord it S0 \<equiv> set_iterator_genord it S0 (\<le>)"
utp_bottom: fixes P :: "'a hrel" assumes "P is \<H>" shows "\<^bold>\<bottom> \<sqsubseteq> P"
hom_grading_varnum_pp: shows "hom_grading varnum_pp"
bind_pmf_cong: fixes p :: "'h pmf"   and q :: "'h pmf"   and f :: "'h \<Rightarrow> 'i pmf"   and g :: "'h \<Rightarrow> 'i pmf" assumes "p = q"   and "\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x" shows "p \<bind> f = q \<bind> g"
ignore_Inl_Inter: fixes S :: "'d set set" assumes "S \<noteq> {}"   and "\<Inter> S \<noteq> {}" shows "(\<Inter>A\<in>S. {(x, y). ignore_Inl x y} `` {x. Basic_BNFs.setl x \<union> Basic_BNFs.setr x \<subseteq> A}) \<subseteq> {(x, y). ignore_Inl x y} `` (\<Inter>A\<in>S. {x. Basic_BNFs.setl x \<union> Basic_BNFs.setr x \<subseteq> A})"
distributed_simple_function_superset: fixes X :: "'a \<Rightarrow> 'e"   and P :: "'e \<Rightarrow> real"   and A :: "'e set" assumes "simple_function M X"   and "\<And>x. x \<in> X ` space M \<Longrightarrow> P x = prob (X -` {x} \<inter> space M)"   and "X ` space M \<subseteq> A"   and "finite A" shows "distributed M (count_space A) X (\<lambda>x. ennreal (if x \<in> X ` space M then P x else 0))"
old_declared: shows "G\<turnstile>Method old declared_in declclass old"
set_remove1_subset: fixes x :: "'c"   and xs :: "'c list" shows "set (remove1 x xs) \<subseteq> set xs"
Odd_lemma: shows "{{Z=Arg+0}. BODY Even .{Res_ok}}|-{Z=Arg+Suc 0}. EvenOdd.odd .{Res_ok}"
pickFT_length: fixes cl :: "('a, 'b, 'c) cmd list" assumes "WtFT cl = 1" shows "pickFT cl < length cl"
fls_inverse_minus: fixes f :: "'a fls" shows "inverse (- f) = - inverse f"
simps_2: fixes x6 :: "ty"   and y6 :: "ty" shows "(x6\<lfloor>\<rceil> = y6\<lfloor>\<rceil>) = (x6 = y6)"
dg_in_ArrE: fixes f :: "V" assumes "f \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>" obtains a :: "V" and b :: "V" where "f : a \<mapsto>\<^bsub>\<BB>\<^esub> b" and "a \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" and "b \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"
eq: shows "f \<equiv> C (option.fixp_fun (\<lambda>f. U (F (C f))))"
wilson_theorem: fixes p :: "nat" assumes "prime p" shows "[fact (p - 1) = - 1] (mod int p)"
more_arith_simps_3: fixes a :: "'a" shows "- a + a = (0::'a)"
altc_cong: fixes return :: "'d \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e"   and altc :: "'f cset \<Rightarrow> ('f \<Rightarrow> 'e) \<Rightarrow> 'e"   and C :: "'f cset"   and f :: "'f \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'e" assumes "monad_altc return bind altc"   and "cBall C (\<lambda>x. f x = g x)" shows "altc C f = altc C g"
mset_permute_list: fixes f :: "nat \<Rightarrow> nat"   and xs :: "'a list" assumes "f permutes {..<length xs}" shows "mset (permute_list f xs) = mset xs"
update_im1: fixes i :: "nat"   and ba :: "'a list \<times> 'a list"   and xs :: "'a list"   and x :: "'a" assumes "list_repr i ba xs"   and "i \<noteq> 0" shows "list_repr i (update_im1 ba x) (xs[i - 1 := x])"
assms: shows "nonneg_cexpr (f -` V) (\<Gamma> \<circ> f) e"
OF_match_linear_ne_Undefined: fixes \<gamma> :: "'b set \<Rightarrow> 'c \<Rightarrow> bool"   and ft :: "('b, 'a) flow_entry_match list"   and p :: "'c" shows "OF_match_linear \<gamma> ft p \<noteq> Undefined"
length_step: fixes s :: "'b list"   and r :: "'b"   and as :: "nat \<times> nat list" shows "length (step s r as) = length s"
Ex1_eq: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "\<exists>!x. P x"   and "P x"   and "P y" shows "x = y"
Rep_process_inverse: fixes x :: "'c process" shows "Abs_process (Rep_process x) = x"
res: shows "split_block A sr1 sc1 = (A1, A2, A3, A4)"
simps_9: fixes t :: "'a local.term"   and u :: "'a local.term"   and v :: "'a local.term" shows "\<^bold>\<lfloor>\<^bold>\<a>\<^bold>[t, u, v\<^bold>]\<^bold>\<rfloor> = (\<^bold>\<lfloor>t\<^bold>\<rfloor> \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> \<^bold>\<lfloor>u\<^bold>\<rfloor>) \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> \<^bold>\<lfloor>v\<^bold>\<rfloor>"
convex_onD: fixes A :: "'c set"   and f :: "'c \<Rightarrow> real"   and t :: "real"   and x :: "'c"   and y :: "'c" assumes "convex_on A f"   and "0 \<le> t"   and "t \<le> 1"   and "x \<in> A"   and "y \<in> A" shows "f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
rel2p_def: fixes R :: "('a \<times> 'b) set"   and x :: "'a"   and y :: "'b" shows "rel2p R x y \<equiv> (x, y) \<in> R"
gr_implies_not0: fixes m :: "nat"   and n :: "nat" assumes "(m::nat) < (n::nat)" shows "(n::nat) \<noteq> 0"
call_no_result_7: shows "all_chains (no_call_to c) \<Gamma> (Rule m' (Call c') # rrs)"
interchange: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "'b \<times> 'b \<Rightarrow> 'b"   and \<alpha> :: "'b \<times> 'b \<times> 'b \<Rightarrow> 'b"   and \<iota> :: "'b"   and h :: "'b"   and g :: "'b"   and h' :: "'b"   and g' :: "'b" assumes "monoidal_category C T \<alpha> \<iota>"   and "partial_magma.arr C (C h g)"   and "partial_magma.arr C (C h' g')" shows "C (T (h, h')) (T (g, g')) = T (C h g, C h' g')"
sinvar_mono_imp_negative_mono: fixes sinvar :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and N :: "'c set"   and E :: "('c \<times> 'c) set"   and E' :: "('c \<times> 'c) set"   and nP :: "'c \<Rightarrow> 'd" assumes "SecurityInvariant_withOffendingFlows.sinvar_mono sinvar"   and "wf_graph \<lparr>nodes = N, edges = E\<rparr>"   and "E' \<subseteq> E"   and "\<not> sinvar \<lparr>nodes = N, edges = E'\<rparr> nP" shows "\<not> sinvar \<lparr>nodes = N, edges = E\<rparr> nP"
exhaust: fixes y :: "'a list" obtains "y = []"   | x21 :: "'a" and x22 :: "'a list" where "y = x21 # x22"
assms: fixes x :: "'a" assumes "x |\<in>| S" shows "P x x"
sub_less_eq_def: fixes x :: "'a"   and y :: "'a" shows "(- x \<le> - y) = (- x \<squnion> - y = - y)"
filter_mset_True: fixes M :: "'b multiset" shows "{#y \<in># M. True#} = M"
below_bottom_iff: fixes x :: "'d" shows "(x \<sqsubseteq> \<bottom>) = (x = \<bottom>)"
cases: fixes \<tau>move :: "('e, 'f, 'c, 'd, 'g, 'h) \<tau>moves"   and a1 :: "(status \<times> 'c) \<times> 'd"   and a2 :: "('e, 'f, status \<times> 'c, 'd, 'g, 'h action) thread_action"   and a3 :: "(status \<times> 'c) \<times> 'd" assumes "\<tau>multithreaded.init_fin_\<tau>move \<tau>move a1 a2 a3" obtains x :: "'c" and m :: "'d" and ta :: "('e, 'f, 'c, 'd, 'g, 'h) thread_action" and x' :: "'c" and m' :: "'d" where "a1 = ((Running, x), m)" and "a2 = convert_TA_initial (convert_obs_initial ta)" and "a3 = ((Running, x'), m')" and "\<tau>move (x, m) ta (x', m')"
monotone_dual: fixes f :: "'a \<Rightarrow> 'a" assumes "Tarski.monotone f (pset cl) (order cl)" shows "Tarski.monotone f (pset (dual cl)) (order (dual cl))"
spanningForest_def: fixes X :: "nat uprod set" shows "spanningForest X = (subforest X \<and> (\<forall>x\<in>E - X. \<not> subforest (insert x X)))"
add_less_imp_less_left: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less"   and "less (plus c a) (plus c b)" shows "less a b"
keys_mapping_of_pmf: fixes p :: "'b pmf" shows "Mapping.keys (mapping_of_pmf p) = set_pmf p"
transaction_receive_def: fixes prot_transaction :: "('f, 'g, 'h, 'e) prot_transaction" shows "transaction_receive prot_transaction = (case prot_transaction of Transaction x1 x2 x3 x4 x5 x6 \<Rightarrow> x2)"
ltrm_pow: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "f \<in> carrier P" shows "monom P ((f [^]\<^bsub>P\<^esub> n) (deg R (f [^]\<^bsub>P\<^esub> n))) (deg R (f [^]\<^bsub>P\<^esub> n)) = monom P (f (deg R f)) (deg R f) [^]\<^bsub>P\<^esub> n"
line_in_subsegment: fixes i1 :: "real"   and i2 :: "real"   and a :: "'b"   and b :: "'b" assumes "i1 < 1"   and "i2 < 1"   and "a \<noteq> b" shows "({a -- b}\<^bsub>i1\<^esub> \<in> {{a -- b}\<^bsub>i2\<^esub><--<b}) = (i2 < i1)"
set_shadow_root_get_disconnected_nodes: fixes ptr :: "(_) element_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) document_ptr" shows "\<forall>w\<in>set_shadow_root_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_disconnected_nodes_locs ptr'. r h h')"
simps: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and C :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and a1 :: "('a, 'c, 'e, 'g) converter"   and a2 :: "('b, 'd, 'f, 'h) converter" shows "rel_converter A B C R a1 a2 = (\<exists>conv1 conv2. a1 = conv1 \<and> a2 = conv2 \<and> rel_fun A (rel_gpv'' (rel_prod B (rel_converter A B C R)) C R) (run_converter conv1) (run_converter conv2))"
ker_subg: fixes F :: "('e, 'f) aGroup_scheme"   and G :: "('g, 'h) aGroup_scheme"   and f :: "'e \<Rightarrow> 'g" assumes "aGroup F"   and "aGroup G"   and "f \<in> aHom F G" shows "F +> ker\<^bsub>F,G\<^esub> f"
inj_on_cmult: fixes c :: "'a" assumes "c \<in> carrier G" shows "inj_on ((\<otimes>) c) (carrier G)"
mk_db_def: shows "mk_db \<equiv> map_fun id Mapping (\<lambda>X p. if p \<in> fst ` X then Some [{ts. (p, ts) \<in> X}] else None)"
set_zip_map1: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "set (zip (map f xs) xs) = (\<lambda>x. (f x, x)) ` set xs"
bij_from_nat: shows "bij_betw mod_type_class.from_nat {0..<CARD('a)} UNIV"
nu_ret: fixes x :: "'a" shows "\<nu> (\<nu> x) = \<nu> x"
fps_mod_def: fixes f :: "'a fps"   and g :: "'a fps" shows "f mod g = (if g = 0 then f else let h = unit_factor g in fps_cutoff (subdegree g) (f * inverse h) * h)"
dim_inverse: fixes x :: "('b, 'a, 'c) Quantity_scheme" shows "Quantity.dim (inverse x) = inverse (Quantity.dim x)"
min_less_self_conv_1: fixes a :: "'a"   and b :: "'a" shows "(min a b < a) = (b < a)"
in_cong_bigtheta: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> \<Theta>[F](g)" shows "(f \<in> L F h) = (g \<in> L F h)"
Catch: shows "Catch r c\<^sub>2 \<in> redexes c"
all_subgroup: shows "\<forall>A\<in>S. subgroup A"
f2: shows "\<forall>s sa sb sc w sd. \<not> low_equal s sa \<or> sb \<noteq> snd (fst (mul_instr_sub1 sc w s)) \<or> sd \<noteq> snd (fst (mul_instr_sub1 sc w sa)) \<or> low_equal sb sd"
fim: shows "f ` sphere 0 1 \<subseteq> S"
invar_qfa: fixes l :: "'b action list" shows "invar (qfa l)"
assms: shows "extend S ( \<LM> At i  \<RM> \<Rightarrow>* \<LM> At i  \<RM>) = ( \<Gamma> \<Rightarrow>* \<Delta>)"
min_le_iff_disj: fixes x :: "'a Word.word"   and y :: "'a Word.word"   and z :: "'a Word.word" shows "(signed.min x y \<le>s z) = (x \<le>s z \<or> y \<le>s z)"
connect_cong_trace: fixes A :: "'a set"   and res1 :: "('a, 'b) resource"   and res2 :: "('a, 'b) resource"   and \<I> :: "('a, 'b) \<I>"   and d :: "('a, 'b) distinguisher" assumes "A \<turnstile>\<^sub>R res1 \<approx> res2"   and "\<I> \<turnstile>g d \<surd>"   and "outs_gpv \<I> d \<subseteq> A"   and "\<I> \<turnstile>res res1 \<surd>"   and "\<I> \<turnstile>res res2 \<surd>" shows "connect d res1 = connect d res2"
s_eq_s_stop: fixes x :: "real" assumes "NO_MATCH other.t_stop x"   and "other.t_stop \<le> x" shows "other.s x = other.s_stop"
Sup_eq_top_iff: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and A :: "'c set" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "(Sup A = top) = (\<forall>x. less x top \<longrightarrow> (\<exists>i\<in>A. less x i))"
vector_idem: fixes x :: "'a" assumes "is_vector x" shows "x ; x = x"
conv_fA: shows "infsetsum'_converges fA (A \<union> B)"
analytic_continuation: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and U :: "complex set"   and \<xi> :: "complex"   and w :: "complex" assumes "f holomorphic_on S"   and "open S"   and "connected S"   and "U \<subseteq> S"   and "\<xi> \<in> S"   and "\<xi> islimpt U"   and "\<And>z. z \<in> U \<Longrightarrow> f z = 0"   and "w \<in> S" shows "f w = 0"
F: shows "offs (enum xs index key n mi ma) 0 ! i \<le> k"
exp_of_derived_img: fixes K :: "'a set"   and n :: "nat" assumes "K \<subseteq> carrier G" shows "(derived H ^^ n) (h ` K) = h ` (derived G ^^ n) K"
discI: fixes option :: "'a option"   and x2 :: "'a" assumes "option = Some x2" shows "option \<noteq> None"
assms_2: shows "(st, st') \<in> \<langle>p2rel (\<langle>nat_rel, fully_unsorted_poly_rel O mset_poly_rel, var_rel\<rangle>pac_step_rel_raw)\<rangle>list_rel"
simple_group_axioms_def: fixes G :: "('e, 'f) monoid_scheme" shows "simple_group_axioms G \<equiv> 1 < order G \<and> (\<forall>H. H \<lhd> G \<longrightarrow> H = carrier G \<or> H = {\<one>\<^bsub>G\<^esub>})"
i1: shows "i < card {i. i < dim_row (A * B) \<and> i \<in> I}"
LeadsTo_Union: fixes S :: "'c set set"   and F :: "'c program"   and B :: "'c set" assumes "\<And>A. A \<in> S \<Longrightarrow> F \<in> A \<longmapsto>w B" shows "F \<in> \<Union> S \<longmapsto>w B"
Return_fair: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> \<box>\<diamond>S1 rmhist p \<longrightarrow> WF(MemReturn memCh (resbar rmhist) p)_(rtrner memCh!p, resbar rmhist!p)"
map_snd_zip_take: fixes xs :: "'e list"   and ys :: "'d list" shows "map snd (xs || ys) = take (min (length xs) (length ys)) ys"
int_lt_neq_prev_lt: fixes a :: "real"   and b :: "real" assumes "a \<in> \<int>"   and "b \<in> \<int>"   and "a - 1 < b"   and "a \<noteq> b" shows "a < b"
simps_3: fixes F :: "'a formula" shows "subformulae (\<^bold>\<not> F) = \<^bold>\<not> F # subformulae F"
indep_in_indep: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E>.indep_in \<E> X" shows "indep_in carrier X"
pivotandupdate_lvars: fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and c :: "'a" assumes "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)" shows "lvars (\<T> (pivot_and_update x\<^sub>i x\<^sub>j c s)) = lvars (\<T> s) - {x\<^sub>i} \<union> {x\<^sub>j}"
fresh_fresh_subst: fixes Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and y :: "'var" assumes "good Y"   and "good X"   and "fresh ys y Y" shows "fresh ys y (X #[Y / y]_ys)"
elims_2: fixes x :: "('a \<times> 'd) list \<times> 'o"   and xa :: "('a \<times> 'd) list \<times> 'o" assumes "oalist_eq_raw x xa" obtains xs :: "('a \<times> 'd) list" and ox :: "'o" and ys :: "('a \<times> 'd) list \<times> 'o" where "x = (xs, ox)" and "xa = ys" and "xs = sort_oalist_aux ox ys"
extend_leadsTo_slice: fixes F :: "'a program"   and AU :: "'c set"   and BU :: "'c set"   and y :: "'b" assumes "extend h F \<in> AU \<longmapsto> BU" shows "F \<in> slice AU y \<longmapsto> project_set h BU"
poly_nth_root_0: fixes n :: "nat"   and p :: "'a poly" assumes "n \<noteq> 0" shows "(poly_nth_root n p = 0) = (p = 0)"