hcju/isabelleps huggingface dataset:
diff_fun_space_order_le: fixes l :: "enat" assumes "l \<le> k" shows "diff_fun_space \<subseteq> c_manifold.diff_fun_space charts l"
fbox_loopI: fixes P :: "'b \<Rightarrow> bool"   and I :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and F :: "'b \<Rightarrow> 'b set" assumes "P \<le> I"   and "I \<le> Q"   and "I \<le> |F] I" shows "P \<le> |LOOP F INV I ] Q"
continuous_map_eq_topcontinuous_at: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" shows "continuous_map X Y f = (\<forall>x\<in>topspace X. topcontinuous_at X Y f x)"
Abs_bit1_inverse: fixes y :: "int" assumes "y \<in> {0..<1 + 2 * int CARD('a)}" shows "Rep_bit1 (Abs_bit1 y) = y"
map_add_le_mapE: fixes f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option"   and h :: "'a \<Rightarrow> 'b option" assumes "f ++ g \<subseteq>\<^sub>m h" shows "g \<subseteq>\<^sub>m h"
set_mset_remdups_mset: fixes A :: "'c multiset" shows "set_mset (remdups_mset A) = set_mset A"
same_level_path_def: fixes as :: "'edge list" shows "same_level_path as \<equiv> same_level_path_aux [] as \<and> upd_cs [] as = []"
le_nat_any_inv: fixes n :: "nat"   and v :: "val" assumes "VNat n \<sqsubseteq> v" obtains "v = VNat n"
prime_factorization_divide: fixes b :: "'a"   and a :: "'a" assumes "b dvd a" shows "prime_factorization (a div b) = prime_factorization a - prime_factorization b"
contractible_imp_simply_connected: fixes S :: "'a set" assumes "contractible S" shows "simply_connected S"
prob_compl: fixes M :: "'b measure"   and A :: "'b set" assumes "prob_space M"   and "A \<in> sets M" shows "Sigma_Algebra.measure M (space M - A) = 1 - Sigma_Algebra.measure M A"
assms_2: shows "rk {B, C, \<alpha>} = 2"
put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok: fixes h :: "(_) heap"   and ptr :: "(_) element_ptr"   and setter :: "('m \<Rightarrow> 'm) \<Rightarrow> ('c, 'a, 'b, 'e, 'g option) RElement_scheme \<Rightarrow> ('c, 'a, 'b, 'e, 'g option) RElement_scheme"   and v :: "'m" assumes "ShadowRootClass.type_wf h"   and "ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr setter v"
util_exactly_one_element: fixes m :: "'d"   and l :: "'d list"   and l' :: "'d list" assumes "m \<notin> set l"   and "l' = l @ [m]" shows "\<exists>!j. j < length l' \<and> l' ! j = m"
roots_of_3_2: fixes p :: "int poly" shows "distinct (map real_of_3 (roots_of_3 p))"
quasi_inverses_are_adjoint_pair: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.quasi_inverses f g" shows "Span.adjoint_pair f g"
2: fixes s1 :: "'a"   and sc1 :: "'a scheduler"   and s2 :: "'a"   and sc2 :: "'a scheduler" assumes "X (s1, sc1) (s2, sc2)" shows "action_sch sc1 = action_sch sc2"
cod_eqI: fixes b :: "'j"   and f :: "'j" assumes "J.ide b"   and "J.seq b f" shows "J.cod f = b"
continuous_closed_graph_gen: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and T :: "'b set" assumes "continuous_on S f"   and "f ` S \<subseteq> T" shows "closedin (top_of_set (S \<times> T)) ((\<lambda>x. (x, f x)) ` S)"
ge_iff_diff_ge_0: fixes b :: "'c"   and a :: "'c" shows "b \<le> a \<equiv> (0::'c) \<le> a - b"
topspace_Euclidean_space_alt: fixes n :: "nat" shows "topspace (Euclidean_space n) = (\<Inter>i\<in>{n..}. {x \<in> topspace (powertop_real UNIV). x i \<in> {0}})"
kruskal_invariant_def: fixes f :: "'a"   and g :: "'a"   and h :: "'a" shows "kruskal_invariant f g h \<equiv> kruskal_spanning_invariant f g h \<and> (\<exists>w. minimum_spanning_forest w g \<and> f \<le> w \<squnion> w\<^sup>T)"
neg_le_0_iff_le: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq (uminus a) zero = less_eq zero a"
pp_dist_comp: fixes x :: "'c"   and y :: "'c" shows "- - (x \<sqinter> y) = - - x \<sqinter> - - y"
unit_in_vhom: fixes a :: "'c" assumes "C.obj a" shows "\<guillemotleft>\<i>\<^sub>C[a] : a \<star>\<^sub>C a \<Rightarrow>\<^sub>C a\<guillemotright>"
cop: shows "comm_monoid_mult_class.coprime (lead_coeff F) p"
assms_1: shows "(k, s) = widest_spread ks (x # y # zs)"
assms_1: shows "monom_mult c t p1 = monom_mult c t p2"
"case": fixes f :: "'h \<Rightarrow> 'i \<Rightarrow> 'g"   and x1 :: "'h"   and x2 :: "'i" shows "(case (x1, x2) of (x, xa) \<Rightarrow> f x xa) = f x1 x2"
type_less_x_Integer_intro: fixes \<tau> :: "'a basic_type" assumes "\<tau> = UnlimitedNatural" shows "\<tau> < Integer"
select_convs_1: fixes dwa_cond :: "'a \<Rightarrow> bool"   and dwa_step :: "'a \<Rightarrow> 'a"   and dwa_initial :: "'a"   and dwa_invar :: "'a set"   and more :: "'b" shows "dwa_cond \<lparr>dwa_cond = dwa_cond, dwa_step = dwa_step, dwa_initial = dwa_initial, dwa_invar = dwa_invar, \<dots> = more\<rparr> = dwa_cond"
assms_5: shows "has_greatest_postfixpoint (h \<circ> u)"
inject_2: fixes x21 :: "dB"   and x22 :: "dB"   and y21 :: "dB"   and y22 :: "dB" shows "(x21 \<degree> x22 = y21 \<degree> y22) = (x21 = y21 \<and> x22 = y22)"
simple_extension_as_eval_img: fixes K :: "'a list set"   and x :: "'a list" assumes "K \<subseteq> carrier (K [X])"   and "x \<in> carrier (K [X])" shows "UP.simple_extension K x = (\<lambda>p. UP.eval p x) ` carrier (K [X]\<^bsub>K [X]\<^esub>)"
sys_xcpts_cases: fixes C :: "String.literal"   and P :: "String.literal \<Rightarrow> bool" assumes "C \<in> sys_xcpts"   and "P NullPointer"   and "P OutOfMemory"   and "P ClassCast"   and "P ArrayIndexOutOfBounds"   and "P ArrayStore"   and "P NegativeArraySize"   and "P ArithmeticException"   and "P IllegalMonitorState"   and "P IllegalThreadState"   and "P InterruptedException" shows "P C"
rot_circ_left_edge_reparam_polar_circ_split: shows "reparam (rec_join [rot_circle_left_edge]) (rec_join [subcube (1 / 4) (1 / 2) (1, circle_polar), subcube (1 / 2) (3 / 4) (1, circle_polar)])"
semiring_norm_16: fixes m :: "num"   and n :: "num" shows "num.Bit1 m * num.Bit1 n = num.Bit1 (m + n + num.Bit0 (m * n))"
ltrm_mult: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "q \<in> carrier P" shows "monom P ((p \<otimes>\<^bsub>P\<^esub> q) (deg R (p \<otimes>\<^bsub>P\<^esub> q))) (deg R (p \<otimes>\<^bsub>P\<^esub> q)) = monom P (p (deg R p)) (deg R p) \<otimes>\<^bsub>P\<^esub> monom P (q (deg R q)) (deg R q)"
assms: fixes i :: "'a" assumes "i \<in> I" shows "Group.group (G i)"
consistencyD: fixes f :: "'a set \<Rightarrow> 'a set"   and C :: "'a set"   and B :: "'a set" assumes "consistency f"   and "C \<subseteq> B"   and "f B \<subseteq> C" shows "f B = f C"
cring_simprules_6: fixes R :: "('a, 'b) ring_scheme" assumes "cring R" shows "\<one>\<^bsub>R\<^esub> \<in> carrier R"
fake_beta: fixes v :: "i"   and \<psi> :: "\<kappa> \<Rightarrow> \<o>"   and x :: "\<nu>" assumes "\<And>G \<phi>. IsProperInX (\<lambda>x. \<lparr>G,\<^bold>\<iota>y. \<phi> y x\<rparr>)" shows "[\<lparr>backdoor \<psi>,x\<^sup>P\<rparr> \<^bold>\<equiv> \<^bold>\<A>\<psi> (x\<^sup>P) in v]"
zero_smult: fixes m :: "'m" assumes "m \<in> M" shows "(0::'r) \<cdot> m = (0::'m)"
simps_2: fixes x :: "'b message_af"   and xs :: "'b message_af list"   and k :: "nat" shows "(x # xs) \<odot> k = (if 0 < k then x # NoMsg\<^bsup>k - Suc 0\<^esup> @ xs \<odot> k else [])"
check_in: fixes s :: "state"   and k :: "key"   and r :: "room"   and g :: "guest" assumes "s \<in> reach"   and "k \<notin> state.issued s" shows "s\<lparr>currk := (state.currk s)(r := k), issued := state.issued s \<union> {k}, cards := (state.cards s)(g := state.cards s g \<union> {(state.currk s r, k)}), owns := (state.owns s)(r := \<lfloor>g\<rfloor>), safe := (state.safe s)(r := False)\<rparr> \<in> reach"
lossless_spmf_inline1: fixes gpv :: "('b, 'call, 'ret) gpv"   and s :: "'s" assumes "lossless_gpv \<I> gpv"   and "\<I> \<turnstile>g gpv \<surd>"   and "\<And>s x. x \<in> outs_\<I> \<I> \<Longrightarrow> lossless_spmf (the_gpv (callee s x))" shows "lossless_spmf (inline1 callee gpv s)"
inject: fixes x1 :: "'b list"   and x2 :: "'b"   and y1 :: "'b list"   and y2 :: "'b" shows "(Infer x1 x2 = Infer y1 y2) = (x1 = y1 \<and> x2 = y2)"
propositional_calculus_16: fixes Q\<^sub>1 :: "fm"   and Q\<^sub>2 :: "fm"   and A :: "fm"   and H :: "fm set"   and C :: "fm"   and B :: "fm" assumes "supp Q\<^sub>1 = {}"   and "supp Q\<^sub>2 = {}"   and "insert A H \<turnstile> C"   and "insert B H \<turnstile> C" shows "insert (A OR B) H \<turnstile> C"
INV_rule: fixes T :: "('a, 'b) TS_scheme"   and I :: "'a set" assumes "init T \<subseteq> I"   and "{I \<inter> reach T} TS.trans T {> I}" shows "reach T \<subseteq> I"
id_eq_obj_3: fixes v :: "i"   and x :: "\<nu>"   and y :: "\<nu>"   and z :: "\<nu>" shows "[x\<^sup>P \<^bold>= y\<^sup>P \<^bold>& y\<^sup>P \<^bold>= z\<^sup>P \<^bold>\<rightarrow> x\<^sup>P \<^bold>= z\<^sup>P in v]"
split_sum_ex: fixes P :: "'d + 'e \<Rightarrow> bool" shows "(\<exists>x. P x) = ((\<exists>x. P (Inl x)) \<or> (\<exists>x. P (Inr x)))"
bij_betw_partitions_of: fixes A :: "'c set"   and B :: "'d set" assumes "finite A"   and "finite B" shows "bij_betw (partitions_of A B) ((A \<rightarrow>\<^sub>E B) // range_permutation A B) {P. partition_on A P \<and> card P \<le> card B}"
intros: fixes q :: "'a"   and f :: "'b"   and qs :: "'a list"   and \<delta> :: "('a, 'b) ta_rule set"   and ts :: "'b tree list" assumes "q \<rightarrow> f qs \<in> \<delta>"   and "length ts = length qs"   and "\<And>i. i < length qs \<Longrightarrow> accs \<delta> (ts ! i) (qs ! i)" shows "accs \<delta> (NODE f ts) q"
scalar_equiv: fixes r :: "real"   and x :: "(real, 'b) vec" shows "r *s x = r *\<^sub>R x"
get_disconnected_document_def: shows "get_disconnected_document \<equiv> l_get_disconnected_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_disconnected_document get_disconnected_nodes"
mono_finite_extension: fixes K :: "'a set"   and K' :: "'a set"   and xs :: "'a list" assumes "K \<subseteq> K'" shows "finite_extension K xs \<subseteq> finite_extension K' xs"
singular_relcycle_def: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set" shows "singular_relcycle p X S \<equiv> \<lambda>c. singular_chain p X c \<and> (chain_boundary p c, 0) \<in> mod_subset (p - 1) (subtopology X S)"
simps: fixes K :: "'a \<Rightarrow> 'a pmf set"   and a1 :: "'a"   and a2 :: "'a cfg" assumes "Markov_Decision_Process K" shows "Markov_Decision_Process.cfg_onp K a1 a2 = (\<exists>cfg s. a1 = s \<and> a2 = cfg \<and> state cfg = s \<and> action cfg \<in> K s \<and> (\<forall>x. x \<in> set_pmf (action cfg) \<longrightarrow> Markov_Decision_Process.cfg_onp K x (cont cfg x)))"
impl_def: fixes a :: "'a G"   and b :: "'a G" shows "a l\<rightarrow> b \<equiv> Abs_G ((Rep_G b - Rep_G a + u) \<sqinter> u)"
simps_3: fixes x2 :: "'d" shows "\<lfloor>x2\<rfloor> \<noteq> \<bottom>"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> k \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f3 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat \<Rightarrow> 'a"   and f4 :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and msg :: "msg" shows "P (case msg of Rreq x xa xb xc xd xe xf \<Rightarrow> f1 x xa xb xc xd xe xf | Rrep x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd | Rerr x xa \<Rightarrow> f3 x xa | Newpkt x xa \<Rightarrow> f4 x xa | Pkt x xa xb \<Rightarrow> f5 x xa xb) = (\<not> ((\<exists>x11 x12 x13 x14 x15 x16 x17. msg = Rreq x11 x12 x13 x14 x15 x16 x17 \<and> \<not> P (f1 x11 x12 x13 x14 x15 x16 x17)) \<or> (\<exists>x21 x22 x23 x24 x25. msg = Rrep x21 x22 x23 x24 x25 \<and> \<not> P (f2 x21 x22 x23 x24 x25)) \<or> (\<exists>x31 x32. msg = Rerr x31 x32 \<and> \<not> P (f3 x31 x32)) \<or> (\<exists>x41 x42. msg = Newpkt x41 x42 \<and> \<not> P (f4 x41 x42)) \<or> (\<exists>x51 x52 x53. msg = Pkt x51 x52 x53 \<and> \<not> P (f5 x51 x52 x53))))"
xyz_eq: shows "{#x, y, z#} = poly_roots [:- 1 / 6, - 1 / 2, - 1, 1:]"
many_reducible_to_via_def: shows "many_reducible_to_via = (\<lambda>A B f. total_recursive f \<and> (\<forall>x. (x \<in> A) = (f x \<in> B)))"
sumlist_Cons: fixes x :: "'a vec"   and xs :: "'a vec list" shows "M.sumlist (x # xs) = x + M.sumlist xs"
sub2: fixes xs :: "'a vec list"   and us :: "'a vec list" shows "rev xs @ snd (sub2_wit xs us) = rev (gram_schmidt_sub n xs us)"
ker_is_submodule: shows "LinearCombinations.submodule K kerT V"
sees_methods_rec: fixes P :: "'b prog"   and C :: "char list"   and D :: "char list"   and fs :: "fdecl list"   and ms :: "'b mdecl list"   and Mm :: "char list \<Rightarrow> ((staticb \<times> ty list \<times> ty \<times> 'b) \<times> char list) option"   and Mm' :: "char list \<Rightarrow> ((staticb \<times> ty list \<times> ty \<times> 'b) \<times> char list) option" assumes "class P C = \<lfloor>(D, fs, ms)\<rfloor>"   and "C \<noteq> Object"   and "P \<turnstile> D sees_methods Mm"   and "Mm' = Mm ++ (map_option (\<lambda>m. (m, C)) \<circ> map_of ms)" shows "P \<turnstile> C sees_methods Mm'"
\<omega>_limit_set_empty: fixes x :: "'a" assumes "\<not> {0..} \<subseteq> existence_ivl0 x" shows "\<omega>_limit_set x = {}"
sum_pmf_eq_1: fixes A :: "'a set"   and p :: "'a pmf" assumes "finite A"   and "set_pmf p \<subseteq> A" shows "sum (pmf p) A = 1"
ts_inv_ok_upd_inv': fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and I :: "'f \<Rightarrow> 'i option"   and ta :: "('f, 'j, 'k) new_thread_action"   and P :: "'i \<Rightarrow> 'f \<Rightarrow> 'j \<Rightarrow> 'k \<Rightarrow> bool" assumes "ts_inv_ok ts I" shows "ts_inv_ok (redT_updT' ts ta) (upd_inv I P ta)"
ind: fixes i :: "nat"   and xs :: "nat list"   and y :: "nat"   and ys :: "nat list" assumes "x = i"   and "length xs = i" shows "godel_code' (xs @ y # ys) (Suc 0) = godel_code' xs (Suc 0) * Pi (Suc i) ^ y * godel_code' ys (Suc (Suc i))"
fst_leaves: fixes t :: "int tree"   and i :: "int"   and j :: "int" assumes "inorder t = [i..j]"   and "i \<le> j + 1" shows "fst ` leaves i t = {i..j + 1}"
coupling_step: fixes Q :: "'v \<Rightarrow> nat option"   and V :: "'v \<Rightarrow> nat option"   and u :: "'v"   and du :: "nat" assumes "coupling Q V D S"   and "u \<notin> S"   and "D u = enat du" shows "coupling ((relax_outgoing' u du V Q)(u := None)) (V(u \<mapsto> du)) (relax_outgoing u D) (Set.insert u S)"
pfinite_id_converter: fixes \<I> :: "('y, 'z) \<I>" shows "pfinite_converter \<I> \<I> 1\<^sub>C"
hom_det: fixes A :: "'a poly mat" shows "det (p.mat_hom A) = map_poly hom (det A)"
lub_eq: fixes below :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and X :: "'e \<Rightarrow> 'd"   and Y :: "'e \<Rightarrow> 'd" assumes "class.cpo below"   and "\<And>i. X i = Y i" shows "po.lub below (range X) = po.lub below (range Y)"
ext_inject: fixes NTDom :: "('g, 'h, 'i, 'j, 'k, 'l) Functor"   and NTCod :: "('g, 'h, 'i, 'j, 'k, 'l) Functor"   and NatTransMap :: "'g \<Rightarrow> 'j"   and more :: "'m"   and NTDom' :: "('g, 'h, 'i, 'j, 'k, 'l) Functor"   and NTCod' :: "('g, 'h, 'i, 'j, 'k, 'l) Functor"   and NatTransMap' :: "'g \<Rightarrow> 'j"   and more' :: "'m" shows "(\<lparr>NTDom = NTDom, NTCod = NTCod, NatTransMap = NatTransMap, \<dots> = more\<rparr> = \<lparr>NTDom = NTDom', NTCod = NTCod', NatTransMap = NatTransMap', \<dots> = more'\<rparr>) = (NTDom = NTDom' \<and> NTCod = NTCod' \<and> NatTransMap = NatTransMap' \<and> more = more')"
size_inter: shows "int (size {#mset_set {y. y \<subseteq> blv \<inter> b2 \<and> card y = 2}. b2 \<in># filter_mset ((\<subseteq>) p) (remove1_mset blv \<B>)#}) = \<Lambda> - 1"
ate_arev: fixes a :: "'b" assumes "a \<in> arcs G" shows "arc_to_ends G (arev a) = prod.swap (arc_to_ends G a)"
swapidseq_ext_backwards': fixes A :: "'a set"   and n :: "nat"   and p :: "'a \<Rightarrow> 'a" assumes "swapidseq_ext A (Suc n) p" shows "\<exists>a b A' p'. a \<in> A \<and> b \<in> A \<and> a \<noteq> b \<and> swapidseq_ext A n p' \<and> p = Fun.swap a b id \<circ> p'"
skolemise: fixes P :: "'a \<Rightarrow> bool" assumes "\<not> All P" shows "\<not> P (SOME x. \<not> P x)"
bl_and_bin: fixes n :: "nat"   and v :: "int"   and w :: "int" shows "map2 (\<and>) (bin_to_bl n v) (bin_to_bl n w) = bin_to_bl n (v AND w)"
False: shows "\<not> (a \<noteq> b \<and> b \<noteq> c \<and> a \<noteq> c)"
content_times_primitive_part: fixes p :: "'a poly" shows "Polynomial.smult (content p) (primitive_part p) = p"
wf_ruleset_fst: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and p :: "'c"   and m :: "'b match_expr"   and a :: "action"   and rs :: "'b rule list" shows "wf_ruleset \<gamma> p (Rule m a # rs) = (wf_ruleset \<gamma> p [Rule m a] \<and> wf_ruleset \<gamma> p rs)"
sylvester_mat_dim_1: fixes p :: "'a poly"   and q :: "'a poly" shows "dim_row (sylvester_mat p q) = degree p + degree q"
elims: fixes x :: "nat"   and xa :: "nat"   and y :: "nat option" assumes "rank x xa = y" obtains xb :: "nat" and n :: "nat" where "x = xb" and "xa = n" and "y = (if xb \<le> n \<and> \<not> sink (token_run xb n) then Some (card (older_seniors xb n)) else None)"
lexord_trans: fixes x :: "'a list"   and y :: "'a list"   and r :: "('a \<times> 'a) set"   and z :: "'a list" assumes "(x, y) \<in> lexord r"   and "(y, z) \<in> lexord r"   and "trans r" shows "(x, z) \<in> lexord r"
subM: fixes t :: "'a set"   and M :: "'a set set" assumes "t \<in> M"   and "M \<subseteq> S" shows "t open"
AGMC_S_axioms: shows "AGMC_S Cn (\<div>) \<top> \<bottom> (.\<longrightarrow>.) .\<not> (.\<and>.) (.\<or>.) (.\<longleftrightarrow>.)"
simps_1: fixes T1 :: "('fsym, 'tp) efsym trm"   and T2 :: "('fsym, 'tp) efsym trm" shows "TE.wtA (Eq T1 T2) = (TE.wt T1 \<and> TE.wt T2 \<and> TE.tpOf T1 = TE.tpOf T2)"
simps_37: fixes vd :: "bexp"   and ve :: "'a acom"   and vf :: "'a acom"   and vg :: "'a"   and v :: "'a"   and va :: "bexp"   and vb :: "'a acom"   and vc :: "'a" shows "(IF vd THEN ve ELSE vf {vg} \<le> {v} WHILE va DO vb {vc}) = False"
inv_locate_a_Bk_via_b: fixes as :: "nat"   and am :: "nat list"   and q :: "nat"   and aaa :: "cell list"   and xs :: "cell list"   and ires :: "cell list" assumes "inv_locate_b (as, am) (q, aaa, Bk # xs) ires"   and "\<nexists>n. xs = Bk \<up> n" shows "inv_locate_a (as, am) (Suc q, Bk # aaa, xs) ires"
assms: shows "GEN_OP un (\<union>) (A \<rightarrow> A \<rightarrow> A)"
simps_1: fixes f :: "'e \<Rightarrow> 'c"   and g :: "'f \<Rightarrow> 'd"   and P :: "'e" shows "chainmap f g \<lbrace> P \<rbrace> = \<lbrace> f P \<rbrace>"
map_poly_0: fixes f :: "'d \<Rightarrow> 'c" shows "map_poly f 0 = 0"
fold_insert: fixes A :: "name set"   and x :: "name"   and z :: "tm" assumes "finite A"   and "x \<notin> A" shows "Finite_Set.fold (\<lambda>i. subst i (F i)) z (insert x A) = subst x (F x) (Finite_Set.fold (\<lambda>i. subst i (F i)) z A)"
compose_Id: fixes g :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "g \<in> A \<rightarrow> B"   and "g \<in> extensional A" shows "compose A g (identity A) = g"
proj_tuple_alt: fixes bs :: "bool list"   and as :: "'a option list" shows "proj_tuple bs as = map2 (\<lambda>x y. if x then y else None) bs as"
gbm_delete_edge_def: fixes v :: "'V"   and e :: "'W"   and v' :: "'V"   and g :: "'m1" shows "gbm_delete_edge v e v' g \<equiv> case m1.lookup v g of None \<Rightarrow> g | Some m2 \<Rightarrow> case m2.lookup v' m2 of None \<Rightarrow> g | Some s3 \<Rightarrow> m1.update v (m2.update v' (s3.delete e s3) m2) g"
f6: shows "(r |\<^bsub>rel\<^esub> s) \<otimes>\<^bsub>rec_rng_of_frac\<^esub> (r'' |\<^bsub>rel\<^esub> s'') \<oplus>\<^bsub>rec_rng_of_frac\<^esub> (r' |\<^bsub>rel\<^esub> s') \<otimes>\<^bsub>rec_rng_of_frac\<^esub> (r'' |\<^bsub>rel\<^esub> s'') = (s' \<otimes> s'' \<otimes> (r \<otimes> r'') \<oplus> s \<otimes> s'' \<otimes> (r' \<otimes> r'') |\<^bsub>rel\<^esub> s \<otimes> s'' \<otimes> (s' \<otimes> s''))"
map_real_pair_def: fixes f :: "real \<Rightarrow> real \<Rightarrow> 'a"   and g :: "val \<Rightarrow> 'a"   and x :: "val" shows "map_real_pair f g x = (case x of <|RealVal a, RealVal b|> \<Rightarrow> f a b | <|RealVal a, _|> \<Rightarrow> g x | <|_, val2|> \<Rightarrow> g x | _ \<Rightarrow> g x)"
eval_fds_def: fixes f :: "'a fds"   and s :: "'a" shows "eval_fds f s = (\<Sum>n. fds_nth f n / nat_power n s)"
fst_Gauss_Jordan_PA: fixes A :: "(('a, 'b) vec, 'c) vec" shows "fst (Gauss_Jordan_PA A) ** A = snd (Gauss_Jordan_PA A)"
leg1_iso: shows "\<exists>\<nu>. \<guillemotleft>\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright> \<and> local.iso \<nu>"
1: shows "w1 = basic (make_vert_block (nat (domain_wall B))) \<circ> B \<and> w2 = B \<circ> basic [] \<and> codomain_wall B = 0 \<and> is_tangle_diagram B"
assms: fixes n :: "nat" assumes "count N x = n + count M x" shows "False"
is_ccompare_def: fixes uu :: "'b itself" shows "is_ccompare uu = (ID ccompare \<noteq> None)"
rank_le_i: shows "rank A \<le> mod_type_class.to_nat i"
add_right_mono: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.ordered_ab_semigroup_add plus less_eq less"   and "less_eq a b" shows "less_eq (plus a c) (plus b c)"
intro: fixes X :: "'b set set" assumes "ChamberComplex X"   and "ThinishChamberComplex_axioms X" shows "ThinishChamberComplex X"
code: fixes x :: "quat"   and y :: "quat" shows "x * y = Quat (quat.Re x * quat.Re y - Im1 x * Im1 y - Im2 x * Im2 y - Im3 x * Im3 y) (quat.Re x * Im1 y + Im1 x * quat.Re y + Im2 x * Im3 y - Im3 x * Im2 y) (quat.Re x * Im2 y - Im1 x * Im3 y + Im2 x * quat.Re y + Im3 x * Im1 y) (quat.Re x * Im3 y + Im1 x * Im2 y - Im2 x * Im1 y + Im3 x * quat.Re y)"
Nmlize_Vcomp_Arr_Dom: fixes t :: "'a local.term" assumes "Arr t" shows "\<^bold>\<lfloor>t \<^bold>\<cdot> Dom t\<^bold>\<rfloor> = \<^bold>\<lfloor>t\<^bold>\<rfloor>"
abs_eq: fixes x :: "\<upsilon> \<Rightarrow> j \<Rightarrow> i \<Rightarrow> bool" shows "d\<^sub>1 (make\<Pi>\<^sub>1 x) = Some x"