hcju/isabelleps huggingface dataset:
select_convs_2: fixes TDomain :: "'a"   and TExprSet :: "('a, 'b) Expression set"   and TCodomain :: "'a"   and more :: "'c" shows "TExprSet \<lparr>TDomain = TDomain, TExprSet = TExprSet, TCodomain = TCodomain, \<dots> = more\<rparr> = TExprSet"
rel_pmf_imp_rel_set: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'b pmf"   and q :: "'c pmf" assumes "rel_pmf R p q" shows "rel_set R (set_pmf p) (set_pmf q)"
9: shows "pd_kleene_allegory_class.acyclic (f\<^sup>T)"
aux: fixes t :: "typ"   and t' :: "typ"   and n :: "nat" assumes "t = t'" shows "subst_to_scheme (\<lambda>k. if n \<le> k then BVar (k - n) else FVar k) t = subst_to_scheme (\<lambda>k. if n \<le> k then BVar (k - n) else FVar k) t'"
assms_3: shows "fst (rep_nat_term v) = fst (rep_nat_term b)"
policy_def: shows "policy \<equiv> \<lparr>nodesL = [1, 2, 3], edgesL = [(1, 2), (2, 2), (2, 3)]\<rparr>"
inject_2: fixes x2 :: "nat"   and y2 :: "nat" shows "(Indet x2 = Indet y2) = (x2 = y2)"
lcm_pos_nat: fixes m :: "nat"   and n :: "nat" assumes "0 < m"   and "0 < n" shows "0 < lcm m n"
aext_bop: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'b"   and u :: "('e, 'd) uexpr"   and v :: "('f, 'd) uexpr"   and a :: "'d \<Longrightarrow> 'c" shows "bop f u v \<oplus>\<^sub>p a = bop f (u \<oplus>\<^sub>p a) (v \<oplus>\<^sub>p a)"
irr: fixes b :: "'a fract poly" assumes "b dvd map_poly to_fract p"   and "\<not> map_poly to_fract p dvd b" shows "is_unit b"
complement_trace_snth: fixes A :: "('c, 'd) nba"   and w :: "'c stream"   and r :: "(('d \<Rightarrow> nat option) \<times> 'd set) stream"   and p :: "('d \<Rightarrow> nat option) \<times> 'd set"   and k :: "nat" assumes "run (complement A) (w ||| r) p" obtains "fst ((p ## trace (w ||| r) p) !! Suc k) \<in> lr_succ A (w !! k) (fst ((p ## trace (w ||| r) p) !! k))" and "snd ((p ## trace (w ||| r) p) !! Suc k) = st_succ A (w !! k) (fst ((p ## trace (w ||| r) p) !! Suc k)) (snd ((p ## trace (w ||| r) p) !! k))"
cod_in_codomains: fixes f :: "'a" assumes "B.codomains f \<noteq> {}" shows "B.cod f \<in> B.codomains f"
length_append: fixes xs :: "'c list"   and ys :: "'c list" shows "length (xs @ ys) = length xs + length ys"
fbox_Q: shows "P \<le> |x\<acute>=f & \<lambda>s. G s \<and> C s on U S @ t\<^sub>0] Q"
pred_set: fixes P :: "'e \<Rightarrow> bool" shows "pred_bset P = (\<lambda>x. Ball (set_bset x) P)"
simps_3: fixes x1 :: "'g" shows "NewEntry \<noteq> Node x1"
sum_up_index_split: fixes f :: "nat \<Rightarrow> 'c"   and m :: "nat"   and n :: "nat" shows "sum f {..m + n} = sum f {..m} + sum f {Suc m..m + n}"
alive_def: fixes x :: "Value"   and s :: "Store" shows "alive x s \<equiv> aliveImpl x (Rep_Store s)"
octo_of_real_sel_5: fixes x :: "real" shows "Im4 (octo_of_real x) = 0"
stone: fixes 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 uminus :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.stone_algebra inf less_eq less sup bot top uminus" shows "sup (uminus x) (uminus (uminus x)) = top"
SUP_eq: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'c set"   and B :: "'d set"   and f :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. less_eq (f i) (g j)"   and "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. less_eq (g j) (f i)" shows "Sup (f ` A) = Sup (g ` B)"
nth_deriv_eq: fixes n :: "nat" shows "(fds_deriv ^^ n) f = fds (\<lambda>k. (- 1) ^ n * fds_nth f k * complex_of_real (ln (real k) ^ n))"
pad2: fixes x :: "'a"   and y :: "'a" shows "- - x \<le> - (- x \<sqinter> y)"
half_open_segment_closed_segmentI: fixes t :: "'a"   and t0 :: "'a"   and t1 :: "'a" assumes "t \<in> {t0--t1}"   and "t \<noteq> t1" shows "t \<in> {t0--<t1}"
wf_digraph_axioms: shows "wf_digraph (app_iso iso_f G)"
strict_trans: fixes b :: "'c multiset"   and a :: "'c multiset"   and c :: "'c multiset" assumes "b \<subset># a"   and "c \<subset># b" shows "c \<subset># a"
mtf2_backwards_effect4: fixes xs :: "'a list"   and q :: "'a"   and n :: "nat"   and i :: "nat" assumes "index xs q < length xs"   and "q \<in> set xs"   and "distinct xs"   and "index (mtf2 n q xs) (xs ! i) < index xs q - n"   and "i < length xs" shows "i < index xs q - n"
white_def: fixes r :: "'h"   and s :: "'i process_name \<Rightarrow> ('j, 'i, 'k, 'h) Model.local_state" shows "white r s \<equiv> obj_at (\<lambda>obj. obj_mark obj \<noteq> sys_fM s) r s"
sup_right_top: fixes x :: "'a" shows "x \<squnion> top = top"
simps_1: fixes x :: "('a, 'b) Combinators" shows "list2policyR [x] = x"
snd2: fixes A :: "'a trm"   and B :: "'a trm" shows "Snd (SimplyTyped.Pair A B) \<rightarrow>\<beta> B"
implies_not_e2: fixes \<Gamma> :: "'c constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and \<Psi> :: "'c TESL_atomic list"   and \<Phi> :: "'c TESL_atomic list" shows "\<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^sub>e K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi>"
gf_F_nonneg: fixes z :: "real"   and x :: "'s"   and y :: "'s" assumes "0 \<le> z"   and "z < 1" shows "0 \<le> gf_F x y z"
hyps_8: fixes b :: "binding list"   and ba :: "binding list"   and bb :: "binding" assumes "\<Delta> @ \<Gamma> = b @ ba"   and "\<turnstile> (b @ bb # ba) ok" shows "b @ bb # ba \<turnstile> t\<^sub>1 : (\<forall>X<:T11. T12)"
strict_trans: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a < b"   and "b < c" shows "a < c"
arr_iff_in_hom: fixes f :: "'c local.term set" shows "\<F>C.arr f = \<guillemotleft>f : \<F>C.dom f \<rightarrow> \<F>C.cod f\<guillemotright>"
simps_2: fixes P :: "'f \<Rightarrow> bool"   and x :: "'f"   and xs :: "'f list" shows "dropWhile P (x # xs) = (if P x then dropWhile P xs else x # xs)"
fl_dghm_op_dghm_is_fl_dghm: shows "op_dghm \<FF> : op_dg \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>u\<^sub>l\<^sub>l\<^bsub>\<alpha>\<^esub> op_dg \<BB>"
pairwise_chain_Union: fixes \<C> :: "'b set set"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>S. S \<in> \<C> \<Longrightarrow> pairwise R S"   and "chain\<^sub>\<subseteq> \<C>" shows "pairwise R (\<Union> \<C>)"
atUV_union_cases: fixes U :: "'a set"   and V :: "'a set"   and c1 :: "'a list multiset"   and c2 :: "'a list multiset" assumes "atUV U V (c1 + c2)" obtains "atUV U V c1"   | "atUV U V c2"   | "atU U c1" and "atU V c2"   | "atU V c1" and "atU U c2"
that: fixes A :: "'a set"   and q :: "'a \<Rightarrow> 'a" assumes "finite A"   and "A \<subseteq> B"   and "\<And>b. q b \<in> range h"   and "p = (\<Sum>b\<in>A. q b * b)" shows "thesis"
obtain_v1_v2_2: shows "v2 \<in> vs' \<union> vs''"
nat_eq_dec: fixes m :: "nat"   and n :: "nat" shows "m = n \<or> m \<noteq> n"
is_process2: fixes P :: "('a event list \<times> 'a event set) set \<times> 'a event list set" assumes "is_process P" shows "\<forall>s X. (s, X) \<in> FAILURES P \<longrightarrow> front_tickFree s"
carrier_is_submodule: shows "Module.submodule (carrier M) R M"
simps_3: fixes T :: "ty"   and U :: "ty" shows "ty_size (T \<otimes> U) = ty_size T + ty_size U + 1"
ccINF_lower: fixes A :: "'c set"   and i :: "'c"   and f :: "'c \<Rightarrow> 'a" assumes "countable A"   and "i \<in> A" shows "Inf (f ` A) \<le> f i"
inf_add_S_right: fixes leq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and add :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and seq_comp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and neutral :: "'c"   and deactivation :: "'c"   and basic :: "'c set"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.stimuli leq add seq_comp neutral deactivation basic"   and "leq x y" shows "leq x (add y z)"
\<alpha>nl_simps_1: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and x :: "('c, 'f) label list" shows "\<alpha>nl fg (ENV x) = \<alpha>n fg x"
Abs_real_int_inverse: fixes y :: "real \<times> real" assumes "y \<in> {r. fst r \<le> snd r}" shows "Rep_real_int (Abs_real_int y) = y"
Diagonal_to_Smith_row_i_PQ_eq: fixes P :: "(('a, 'c) vec, 'c) vec"   and A :: "(('a, 'b) vec, 'c) vec"   and Q :: "(('a, 'b) vec, 'b) vec"   and B :: "(('a, 'b) vec, 'c) vec"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and P' :: "(('a, 'c) vec, 'c) vec"   and B' :: "(('a, 'b) vec, 'c) vec"   and Q' :: "(('a, 'b) vec, 'b) vec"   and i :: "nat" assumes "P ** A ** Q = B"   and "invertible P"   and "invertible Q"   and "is_bezout_ext bezout"   and "(P', B', Q') = Diagonal_to_Smith_row_i_PQ i bezout (P, B, Q)"   and "isDiagonal B"   and "i < min (nrows A) (ncols A)" shows "B' = Diagonal_to_Smith_row_i B i bezout"
subst_subst: fixes t :: "('d, 'g) Term.term"   and \<sigma> :: "'g \<Rightarrow> ('d, 'f) Term.term"   and \<tau> :: "'f \<Rightarrow> ('d, 'e) Term.term" shows "t \<cdot> \<sigma> \<cdot> \<tau> = t \<cdot> \<sigma> \<circ>\<^sub>s \<tau>"
safe_regex_map_regex: fixes r :: "'a Regex.regex"   and safe :: "safety \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a"   and fv :: "'a \<Rightarrow> 'b set"   and m :: "modality"   and g :: "safety" assumes "\<And>g x. \<lbrakk>x \<in> regex.atms r; safe g x\<rbrakk> \<Longrightarrow> safe g (f x)"   and "\<And>x. x \<in> regex.atms r \<Longrightarrow> fv (f x) = fv x"   and "Regex.safe_regex fv safe m g r" shows "Regex.safe_regex fv safe m g (regex.map_regex f r)"
subset_Pow_Union: fixes A :: "'b set set" shows "A \<subseteq> Pow (\<Union> A)"
wf_graph_remove_edges: fixes V :: "'b set"   and E :: "('b \<times> 'b) set"   and X :: "('b \<times> 'b) set" assumes "wf_graph \<lparr>nodes = V, edges = E\<rparr>" shows "wf_graph \<lparr>nodes = V, edges = E - X\<rparr>"
shift_ms_aux_def: fixes e :: "real"   and F :: "('b \<times> real) msllist" shows "shift_ms_aux e F = scale_shift_ms_aux (const_expansion 1, e) F"
eq_iff: fixes x :: "'b poly"   and y :: "'b poly" shows "(map_poly to_fract x = map_poly to_fract y) = (x = y)"
Transset_Vset: fixes i :: "V" shows "Transset (Vset i)"
in_lset_ldropWhileD: fixes x :: "'c"   and P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" assumes "x \<in> lset (ldropWhile P xs)" shows "x \<in> lset xs"
minPath_def: shows "(ARG_MIN length io. io \<in> {p'. path M p' q1 \<and> target p' q1 = q2 \<and> length p' \<le> length p}) \<in> {p'. path M p' q1 \<and> target p' q1 = q2 \<and> length p' \<le> length p} \<and> (\<forall>p'\<in>{p'. path M p' q1 \<and> target p' q1 = q2 \<and> length p' \<le> length p}. length (ARG_MIN length io. io \<in> {p'. path M p' q1 \<and> target p' q1 = q2 \<and> length p' \<le> length p}) \<le> length p')"
poly_ext: fixes p :: "'a poly"   and q :: "'a poly" assumes "\<And>x. poly p x = poly q x" shows "p = q"
ext_inject: fixes imethods :: "(sig \<times> mhead) list"   and more :: "'b"   and imethods' :: "(sig \<times> mhead) list"   and more' :: "'b" shows "(\<lparr>imethods = imethods, \<dots> = more\<rparr> = \<lparr>imethods = imethods', \<dots> = more'\<rparr>) = (imethods = imethods' \<and> more = more')"
primeideal_axioms_def: fixes I :: "'a set"   and R :: "('a, 'b) ring_scheme" shows "primeideal_axioms I R \<equiv> carrier R \<noteq> I \<and> (\<forall>a b. a \<in> carrier R \<longrightarrow> b \<in> carrier R \<longrightarrow> a \<otimes>\<^bsub>R\<^esub> b \<in> I \<longrightarrow> a \<in> I \<or> b \<in> I)"
less_irrefl: fixes ko :: "'b key_order"   and x :: "'b" shows "\<not> ko.lt ko x x"
R_2: fixes nb :: "nat" shows "Sup (emeasure (P' nb) ` {K. K \<subseteq> fm nb ` B nb \<and> compact K}) \<noteq> \<infinity>"
has_vderiv_on_compose: fixes f :: "real \<Rightarrow> 'b"   and f' :: "real \<Rightarrow> 'b"   and g :: "real \<Rightarrow> real"   and T :: "real set"   and g' :: "real \<Rightarrow> real" assumes " D f = f' on g ` T"   and " D g = g' on T" shows " D (f \<circ> g) = (\<lambda>x. g' x *\<^sub>R f' (g x)) on T"
length_plus_iarray: fixes A :: "'a iarray"   and B :: "'a iarray" shows "IArray.length (A + B) = max (IArray.length A) (IArray.length B)"
simps_13: fixes t :: "Syntax.term"   and f :: "(Syntax.var, Syntax.term) fmap" shows "psubst_term (Prj2 t) f = Prj2 (psubst_term t f)"
card_of_card_order_on: fixes A :: "'c set" shows "card_order_on A |A|"
prev_Some_less: fixes j :: "nat"   and i :: "nat"   and j' :: "nat"   and i' :: "nat" assumes "prev j i = Some (j', i')"   and "i \<le> j" shows "i' < j'"
that: fixes T :: "real"   and n :: "'a" assumes "0 < T"   and "n = a - b \<or> n = b - a"   and "\<And>x. x \<in> {a--b} \<Longrightarrow> {- T..T} \<subseteq> existence_ivl0 x"   and "\<And>x s. \<lbrakk>x \<in> {a--b}; 0 < s; s \<le> T\<rbrakk> \<Longrightarrow> 0 < (flow0 x s - x) \<bullet> rot n"   and "\<And>x s. \<lbrakk>x \<in> {a--b}; - T \<le> s; s < 0\<rbrakk> \<Longrightarrow> (flow0 x s - x) \<bullet> rot n < 0" shows "thesis"
insert_1: shows "\<forall>L1 L2. finite L1 \<longrightarrow> (\<forall>l\<in>dom F1. \<forall>s p. s \<notin> L1 \<and> p \<notin> L1 \<and> s \<noteq> p \<longrightarrow> (\<exists>t. (the (F1 l)\<^bsup>[Fvar s,Fvar p]\<^esup> => t \<and> (\<forall>z. the (F1 l)\<^bsup>[Fvar s,Fvar p]\<^esup> => z \<longrightarrow> (\<exists>u. t => u \<and> z => u))) \<and> the (F3 l) = \<sigma>[s,p] t)) \<longrightarrow> finite L2 \<longrightarrow> (\<forall>l\<in>dom F1. \<forall>s p. s \<notin> L2 \<and> p \<notin> L2 \<and> s \<noteq> p \<longrightarrow> (\<exists>t. the (F1 l)\<^bsup>[Fvar s,Fvar p]\<^esup> => t \<and> the (F2 l) = \<sigma>[s,p] t)) \<longrightarrow> (\<exists>L'. finite L' \<and> (\<exists>lu. dom lu = dom F1 \<and> (\<forall>l\<in>dom F1. \<forall>s p. s \<notin> L' \<and> p \<notin> L' \<and> s \<noteq> p \<longrightarrow> (\<exists>t. the (F3 l)\<^bsup>[Fvar s,Fvar p]\<^esup> => t \<and> the (lu l) = \<sigma>[s,p] t)) \<and> (\<forall>l\<in>dom F1. body (the (F3 l))) \<and> (\<forall>l\<in>dom F1. \<forall>s p. s \<notin> L' \<and> p \<notin> L' \<and> s \<noteq> p \<longrightarrow> (\<exists>t. the (F2 l)\<^bsup>[Fvar s,Fvar p]\<^esup> => t \<and> the (lu l) = \<sigma>[s,p] t)) \<and> (\<forall>l\<in>dom F1. body (the (F2 l)))))"
nonneg_vel_other: shows "0 \<le> v\<^sub>o"
lmeasurable_interior: fixes S :: "'c set" assumes "bounded S" shows "interior S \<in> lmeasurable"
vars\<^sub>s\<^sub>s\<^sub>t_def: fixes S :: "('f, 'e) stateful_strand_step list" shows "vars\<^sub>s\<^sub>s\<^sub>t S \<equiv> \<Union> (set (map vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p S))"
Fun_mkIde: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and A :: "'b set" assumes "set_category_given_img S img \<AA>"   and "A \<subseteq> Collect (category.terminal S)"   and "|A| <o \<AA>" shows "set_category_given_img.Fun S img (set_category_given_img.mkIde S img \<AA> A) = (\<lambda>x\<in>A. x)"
cmp_props_1: shows "\<guillemotleft>cmp : src \<rho>\<sigma>.tab \<rightarrow> src tab\<guillemotright>"
gr_implies_not_zero: fixes m :: "'b multiset"   and n :: "'b multiset" assumes "m \<subset># n" shows "n \<noteq> {#}"
assms_2: shows "[b] \<preceq>\<^sub>F\<^sub>I w"
elims_1: fixes x :: "cmd"   and xa :: "w_node LDCFG_node"   and xb :: "cmd"   and y :: "bool" assumes "labels_LDCFG_nodes x xa xb = y" obtains prog :: "cmd" and n :: "w_node" and c :: "cmd" where "x = proga__" and "xa = LDCFG_node.Node n" and "xb = c__" and "y = labels_nodes proga__ n c__"   | prog :: "cmd" and c :: "cmd" where "x = progb__" and "xa = NewEntry" and "xb = ca__" and "\<not> y"   | prog :: "cmd" and c :: "cmd" where "x = prog" and "xa = NewExit" and "xb = c" and "\<not> y"
pair_of_term_injective: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and u :: "'a"   and v :: "'a" assumes "term_powerprod pair_of_term term_of_pair"   and "pair_of_term u = pair_of_term v" shows "u = v"
order_le_neq_trans: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b"   and "a \<noteq> b" shows "a < b"
main_lemma: fixes PROB :: "(('c, bool) fmap \<times> ('c, bool) fmap) set"   and s :: "('c, bool) fmap"   and as :: "(('c, bool) fmap \<times> ('c, bool) fmap) list" assumes "finite PROB"   and "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB" shows "\<exists>as'. exec_plan s as = exec_plan s as' \<and> subseq as' as \<and> length as' \<le> 2 ^ card (prob_dom PROB) - 1"
hcomp_simps_2: fixes \<nu> :: "'b"   and \<mu> :: "'b" assumes "D.hseq \<nu> \<mu>" shows "trg\<^sub>D (\<nu> \<star>\<^sub>D \<mu>) = trg\<^sub>D \<nu>"
segment_split: fixes y :: "'c"   and f :: "'c \<Rightarrow> 'c"   and x :: "'c"   and z :: "'c" assumes "y \<in> segment f x z" shows "segment f x z = segment f x y \<union> {y} \<union> segment f y z"
closedin_INT: fixes A :: "'c set"   and U :: "'d topology"   and B :: "'c \<Rightarrow> 'd set" assumes "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> closedin U (B x)" shows "closedin U (\<Inter> (B ` A))"
vfsequence_vdomain: shows "\<D>\<^sub>\<circ> \<AA> = vcard \<AA>"
surjective_respects_domain_permutation: fixes A :: "'c set"   and B :: "'d set" shows "(\<lambda>f. f ` A = B) respects domain_permutation A B"
assms_4: shows "Regions.valid_region \<X> k I r"
mat_of_cblinfun_id: shows "mat_of_cblinfun id_cblinfun = 1\<^sub>m (length canonical_basis)"
assms_2: shows "x\<^sub>i \<in> lvars (\<T> s)"
mkCone_def: fixes e :: "'a" shows "D.mkCone e \<equiv> \<lambda>j. if J.arr j then if j = J.Zero then e else f0 \<cdot> e else null"
permute_minus_cancel_2: fixes permute :: "perm \<Rightarrow> 'a \<Rightarrow> 'a"   and p :: "perm"   and x :: "'a" assumes "class.pt permute" shows "permute (- p) (permute p x) = x"
holo2: shows "(\<lambda>z. CLBINT u=ereal (exp (- 1))..1. complex_of_real (- ln u) powr z / complex_of_real (1 - u) * complex_of_real (- ln u + u - 1)) holomorphic_on {z. - 2 < Re z}"
msub_mOp: fixes M :: "('e, 'a, 'f) Module_scheme"   and M1 :: "('e, 'a, 'f) Module_scheme"   and x :: "'e" assumes "R module M"   and "R module M1"   and "msubmodule R M M1"   and "x \<in> carrier M1" shows "-\<^sub>a\<^bsub>M1\<^esub> x = -\<^sub>a\<^bsub>M\<^esub> x"
gen_subalgebra_space: fixes M :: "'c measure"   and G :: "'c set set" shows "space (gen_subalgebra M G) = space M"
analytic_on_Union: fixes f :: "complex \<Rightarrow> complex"   and \<T> :: "complex set set" shows "(f analytic_on \<Union> \<T>) = (\<forall>T\<in>\<T>. f analytic_on T)"
decomp_vars_3: fixes t :: "('fun, 'var) Term.term" shows "bvars\<^sub>s\<^sub>t (decomp t) = {}"
timedelayed_e2: fixes \<Gamma> :: "'b constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and \<delta>\<tau> :: "'b tag_const"   and K\<^sub>2 :: "clock"   and K\<^sub>3 :: "clock"   and \<Psi> :: "'b TESL_atomic list"   and \<Phi> :: "'b TESL_atomic list" shows "\<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^sub>e K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi>"
lm008: fixes P :: "('c \<times> 'd) set"   and Q :: "('c \<times> 'd) set" shows "P +< Q = Outside' (Domain Q) P +< Q"
ABORT_neq_DONE: fixes a :: "'f" shows "ABORT \<noteq> DONE a"
simps_8: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and P :: "String.literal \<Rightarrow> nat option"   and \<phi> :: "Formula.formula"   and \<psi> :: "Formula.formula"   and j :: "nat" shows "Monitor.progress \<sigma> P (formula.And \<phi> \<psi>) j = min (Monitor.progress \<sigma> P \<phi> j) (Monitor.progress \<sigma> P \<psi> j)"
Cons_1: fixes state :: "State"   and newWl :: "nat list" assumes "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "\<forall>c. c \<in> set Wl' \<longrightarrow> 0 \<le> c \<and> c < length (getF state)"   and "\<forall>c l. l \<noteq> literal \<longrightarrow> (c \<in> set (getWatchList state l)) = (Some l = getWatch1 state c \<or> Some l = getWatch2 state c)"   and "\<forall>c. (c \<in> set newWl \<or> c \<in> set Wl') = (Some literal = getWatch1 state c \<or> Some literal = getWatch2 state c)"   and "set Wl' \<inter> set newWl = {}"   and "uniq Wl'"   and "uniq newWl" shows "let state' = notifyWatches_loop literal Wl' newWl state in InvariantWatchListsCharacterization (getWatchList state') (getWatch1 state') (getWatch2 state') \<and> InvariantWatchListsUniq (getWatchList state')"
generate_pow_on_finite_carrier: fixes a :: "'a" assumes "finite (carrier (mult_of R))"   and "a \<in> carrier (mult_of R)" shows "generate (mult_of R) {a} = {a [^] k |k. k \<in> UNIV}"
wf_cdecl_def: fixes wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" shows "wf_cdecl wf_md P \<equiv> \<lambda>(C, Bs, fs, ms). (\<forall>M mthd Cs. P \<turnstile> C has M = mthd via Cs \<longrightarrow> (\<exists>mthd' Cs'. P \<turnstile> (C, Cs) has overrider M = mthd' via Cs')) \<and> (\<forall>f\<in>set fs. wf_fdecl P f) \<and> distinct_fst fs \<and> (\<forall>m\<in>set ms. wf_mdecl wf_md P C m) \<and> distinct_fst ms \<and> (\<forall>D\<in>baseClasses Bs. is_class P D \<and> (D, C) \<notin> (subcls1 P)\<^sup>* \<and> (\<forall>(M, Ts, T, m)\<in>set ms. \<forall>Ts' T' m' Cs. P \<turnstile> D has M = (Ts', T', m') via Cs \<longrightarrow> Ts' = Ts \<and> P \<turnstile> T \<le> T'))"
z_lastocc: shows "\<forall>j<i - k - 1. cs2 ! j \<noteq> cs ! k"
obj_def: fixes a :: "'a" shows "B.obj a \<equiv> B.arr a \<and> trg a = a"
Sup_inter_less_eq: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "less_eq (Sup (A \<inter> B)) (inf (Sup A) (Sup B))"
IH1_3: fixes i :: "nat" assumes "i < Suc p" shows "xs'' ! i = xs' ! i"
paths_le_pointwise: fixes k :: "nat" shows "pointwise (paths_le k)"
simps: fixes \<Sigma> :: "exesignature"   and ax :: "Core.term list" shows "translate_theory (ExeTheory \<Sigma> ax) = (if exe_sig_conds \<Sigma> then (translate_signature \<Sigma>, set ax) else ((Map.empty, Map.empty, {}, [STR ''A'' \<mapsto> [STR ''A'' \<mapsto> [insert STR ''A'' full_sort]]]), {}))"
similar_mat_wit_sym: fixes A :: "'a mat"   and B :: "'a mat"   and P :: "'a mat"   and Q :: "'a mat" assumes "similar_mat_wit A B P Q" shows "similar_mat_wit B A Q P"
dgrad_sig_set'_def: fixes j :: "nat"   and d :: "'a \<Rightarrow> nat" shows "dgrad_sig_set' j d = dgrad_max_set d \<inter> sig_inv_set' j"
UP_root_div_closed: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "f \<in> carrier P"   and "a \<in> carrier R" shows "UP_root_div f a \<in> carrier P"
solve_depressed_cubic_Cardano_real_def: fixes e :: "real"   and f :: "real" shows "solve_depressed_cubic_Cardano_real e f = (if e = 0 then Some (root 3 (- f)) else let v = - (e ^ 3 / 27) in case rroots2 [:v, f, 1:] of [] \<Rightarrow> None | [u] \<Rightarrow> None | [u, x] \<Rightarrow> let rt = root 3 u in Some (rt - e / (3 * rt)) | u # x # ab # xa \<Rightarrow> Map.empty xa)"