hcju/isabelleps huggingface dataset:
simps_52: fixes f1 :: "('d, 'e) Term.term \<Rightarrow> 'c"   and f2 :: "('d, 'e) Term.term \<Rightarrow> 'c"   and f3 :: "poscheckvariant \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f4 :: "('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f5 :: "('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f6 :: "poscheckvariant \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f7 :: "'e list \<Rightarrow> (('d, 'e) Term.term \<times> ('d, 'e) Term.term) list \<Rightarrow> (('d, 'e) Term.term \<times> ('d, 'e) Term.term) list \<Rightarrow> 'c"   and x31 :: "poscheckvariant"   and x32 :: "('d, 'e) Term.term"   and x33 :: "('d, 'e) Term.term" shows "(case \<langle>x31: x32 \<doteq> x33\<rangle> of send\<langle>x\<rangle> \<Rightarrow> f1 x | receive\<langle>x\<rangle> \<Rightarrow> f2 x | \<langle>x: xa \<doteq> xb\<rangle> \<Rightarrow> f3 x xa xb | insert\<langle>x,xa\<rangle> \<Rightarrow> f4 x xa | delete\<langle>x,xa\<rangle> \<Rightarrow> f5 x xa | \<langle>x: xa \<in> xb\<rangle> \<Rightarrow> f6 x xa xb | \<forall>x\<langle>\<or>\<noteq>: xa \<or>\<notin>: xb\<rangle> \<Rightarrow> f7 x xa xb) = f3 x31 x32 x33"
prems_8: shows "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) M"
y: shows "y \<in> {real_of_float l2..real_of_float u2}"
nat_pow_mult: fixes G :: "('d, 'e) monoid_scheme"   and x :: "'d"   and n :: "nat"   and m :: "nat" assumes "Group.monoid G"   and "x \<in> carrier G" shows "x [^]\<^bsub>G\<^esub> n \<otimes>\<^bsub>G\<^esub> x [^]\<^bsub>G\<^esub> m = x [^]\<^bsub>G\<^esub> (n + m)"
bdd_below_Ici: fixes a :: "'a" shows "bdd_below {a..}"
get_offending_flows_subseteq_edges: fixes M :: "'b SecurityInvariant_configured list"   and F :: "('b \<times> 'b) set"   and V :: "'b set"   and E :: "('b \<times> 'b) set" assumes "valid_reqs M"   and "F \<in> get_offending_flows M \<lparr>nodes = V, edges = E\<rparr>" shows "F \<subseteq> E"
while_hoare_r: fixes p :: "'a upred"   and b :: "'a upred"   and S :: "'a hrel" assumes "\<lbrace>p \<and> b\<rbrace> S \<lbrace>p\<rbrace>\<^sub>u" shows "\<lbrace>p\<rbrace> while\<^sup>\<top> b do S od \<lbrace>\<not> b \<and> p\<rbrace>\<^sub>u"
lhd_conv_lnth: fixes xs :: "'b llist" assumes "\<not> lnull xs" shows "lhd xs = xs ?! 0"
exec_step_ind_Invoke_Null: fixes stk :: "val list"   and n :: "nat"   and M :: "char list"   and P :: "jvm_prog"   and h :: "heap"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" assumes "stk ! n = Null" shows "exec_step_ind (StepI (Invoke M n)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (\<lfloor>addr_of_sys_xcpt NullPointer\<rfloor>, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, ics) # frs, sh)"
IHAPP_5: shows "\<alpha>ah (map (\<alpha>nl fg) w) = h"
canonic_int_validates_all_clauses: fixes S :: "'at Literal set set"   and C :: "'at Literal set" assumes "saturated_binary_rule ordered_resolvent S"   and "all_fulfill finite S"   and "{} \<notin> S"   and "C \<in> S" shows "canonic_int_ordered S \<Turnstile> C"
subgraph_app_isoI': fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism"   and H :: "('a, 'b) pre_digraph"   and H' :: "('a, 'b) pre_digraph" assumes "digraph_isomorphism hom"   and "subgraph H H'"   and "subgraph H' G" shows "subgraph (app_iso hom H) (app_iso hom H')"
is_aligned_mask: fixes w :: "'a word"   and n :: "nat" shows "is_aligned w n = (w && mask n = 0)"
ring_of_sets_axioms: shows "ring_of_sets \<Omega> M"
noFaultn_startD: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "'e"   and f :: "'f" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> Normal t" shows "s \<noteq> Fault f"
mod_sub_add: fixes k :: "nat"   and a :: "nat"   and m :: "nat"   and b :: "nat" assumes "k \<le> a" shows "((a - k) mod m = b mod m) = (a mod m = (b + k) mod m)"
isGlbD3: fixes R :: "'a set"   and S :: "'a set"   and x :: "'a" assumes "isGlb R S x" shows "greatestP (isLb R S) x"
simps_1: fixes ys :: "'b list" shows "shuffles [] ys = {ys}"
that: fixes ys' :: "'a list" assumes "ys' \<in> listset xs'"   and "length ys' = length ys"   and "\<And>i. i < length ys \<Longrightarrow> ys' ! i = ys ! f i" shows "thesis"
simps_2: fixes s :: "'e"   and p :: "'f \<times> 'e"   and ps :: "('f \<times> 'e) list" shows "last_state (s, Cons p ps) = snd p"
simple_match_to_of_matchD: fixes gr :: "of_match_field set"   and r :: "32 simple_match"   and ifs :: "char list list"   and p :: "(32, 'b) simple_packet_ext_scheme" assumes "gr \<in> set (simple_match_to_of_match r ifs)"   and "OF_match_fields gr p = Some True"   and "match_iface (oiface r) (p_oiface p)"   and "simple_match_valid r" shows "simple_matches r p"
less_real_int_def: fixes r :: "real_int"   and s :: "real_int" shows "r < s \<equiv> left s \<le> left r \<and> right r \<le> right s \<and> \<not> (left r \<le> left s \<and> right s \<le> right r)"
subst_apply_fv_empty: fixes t :: "('f, 'e) Term.term"   and \<sigma> :: "'e \<Rightarrow> ('f, 'g) Term.term" assumes "fv t = {}" shows "fv (t \<cdot> \<sigma>) = {}"
EndPhase0_HInv3: fixes s :: "state"   and s' :: "state"   and t :: "Proc"   and p :: "Proc"   and q :: "Proc"   and d :: "Disk" assumes "EndPhase0 s s' t"   and "HInv3_inner s p q d" shows "HInv3_inner s' p q d"
unit_in_hom_2: shows "\<guillemotleft>\<eta>\<^sub>1 : src e\<^sub>1 \<Rightarrow> d\<^sub>1 \<star> e\<^sub>1\<guillemotright>"
ptensor_encode1_encode2: shows "partial_state.encode1 d.dims0 d.vars1' = partial_state.encode2 d.dims0 d.vars2'"
bot_fun_def: shows "\<bottom> = (\<lambda>x. \<bottom>)"
fun_upd_def: fixes f :: "'e \<Rightarrow> 'f"   and a :: "'e"   and b :: "'f" shows "f(a := b) = (\<lambda>x. if x = a then b else f x)"
dsf: fixes k :: "nat" assumes "k < n" shows "matrix_sum d f k \<in> carrier_mat d d"
unsynlr_bot: shows "unsynlr bot = {\<bottom>} \<times> UNIV"
map_add_restrict_comp_right: fixes m :: "'c \<Rightarrow> 'd option"   and S :: "'c set" shows "m |` S ++ m |` (UNIV - S) = m"
vector_smult_rzero: fixes c :: "'c" shows "c *s 0 = 0"
concat_in_hull': fixes ws :: "'b list list"   and G :: "'b list set" assumes "ws \<in> lists G" shows "concat ws \<in> \<langle>G\<rangle>"
ring_distribs_1: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a * (b + c) = a * b + a * c"
perfect_hiding: fixes \<eta> :: "nat"   and \<A> :: "('grp \<times> nat \<Rightarrow> ((nat \<times> nat) \<times> 'b) spmf) \<times> ('grp \<Rightarrow> 'b \<Rightarrow> bool spmf)" shows "Schnorr_\<Sigma>_commit.abstract_com.perfect_hiding_ind_cpa \<eta> \<A>"
CAS_def: shows "CAS = {CAs \<in> lists (CC \<union> {C}). length CAs \<le> max_ary}"
collinearity2: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a"   and e :: "'a"   and de :: "'a set" assumes "\<triangle> a b c"   and "[[b c d]]"   and "[[c e a]]"   and "path de d e" shows "\<exists>f\<in>de. [[a f b]] \<and> [[d e f]]"
path_from_toE: fixes v :: "'a"   and xs :: "'a list"   and w :: "'a" assumes "v \<leadsto>xs\<leadsto>\<^bsub>G\<^esub> w" shows "G.path xs \<and> xs \<noteq> [] \<and> hd xs = v \<and> last xs = w"
add_aform'_float_2: fixes p :: "nat"   and x :: "(real \<times> real pdevs) \<times> real"   and y :: "(real \<times> real pdevs) \<times> real"   and a :: "real"   and b :: "real pdevs"   and ba :: "real" assumes "add_aform' p x y = ((a, b), ba)" shows "ba \<in> float"
assms: shows "(ys, t) \<in> set (incs a x (xs, s))"
word_eq_numeral_iff_iszero_11: fixes x :: "num" shows "(- numeral x = 0) = iszero (numeral x)"
rel_option_restrict_relpE: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool"   and x :: "'c option"   and y :: "'d option" assumes "rel_option (R \<upharpoonleft> P \<otimes> Q) x y" obtains "rel_option R x y" and "pred_option P x" and "pred_option Q y"
reaches_imp_states: fixes fsm :: "'b fsm"   and st :: "hf"   and xs :: "'b list"   and st' :: "hf" assumes "reaches fsm st xs st'" shows "st \<^bold>\<in> states fsm \<and> st' \<^bold>\<in> states fsm"
T_vrestr_0_2: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and A :: "'b set" assumes "qmpt M T"   and "A \<in> sets M" shows "qmpt.vimage_restr M id A = A"
short_vector_external_def: fixes \<alpha> :: "rat"   and fs :: "int vec list" shows "short_vector_external \<alpha> fs = hd (reduce_basis_external \<alpha> fs)"
triangle_in_hom_1: shows "\<guillemotleft>(\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, f\<^sup>*, f] \<cdot> (f \<star> \<eta>) : f \<star> src f \<Rightarrow> trg f \<star> f\<guillemotright>"
saturated_def: fixes N :: "'f set" shows "saturated N = (Inf_from N \<subseteq> Red_I N)"
round_mono: fixes x :: "'b"   and y :: "'b" assumes "x \<le> y" shows "round x \<le> round y"
him_minus: fixes q :: "quat" shows "HIm (- q) = - HIm q"
cfun_map_beta: fixes a :: "'d \<rightarrow> 'e"   and b :: "'f \<rightarrow> 'c"   and f :: "'e \<rightarrow> 'f"   and x :: "'d" shows "cfun_map\<cdot>a\<cdot>b\<cdot>f\<cdot>x = b\<cdot>(f\<cdot>(a\<cdot>x))"
vbrelation_Limit_vsubset_VsetI: fixes \<alpha> :: "V" assumes "Limit \<alpha>"   and "\<D>\<^sub>\<circ> (\<CC>\<lparr>Cod\<rparr>) \<subseteq>\<^sub>\<circ> Vset \<alpha>"   and "\<R>\<^sub>\<circ> (\<CC>\<lparr>Cod\<rparr>) \<subseteq>\<^sub>\<circ> Vset \<alpha>" shows "\<CC>\<lparr>Cod\<rparr> \<subseteq>\<^sub>\<circ> Vset \<alpha>"
isMaxFlow_alt: fixes f :: "nat \<times> nat \<Rightarrow> 'a" shows "isMaxFlow f = (NFlow c s t f \<and> (\<forall>f'. NFlow c s t f' \<longrightarrow> Flow.val c s f' \<le> Flow.val c s f))"
WhileInt: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and b :: "'e set"   and c :: "('e, 'd, 'f) com"   and A :: "'e set"   and V :: "('e \<times> 'e) set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P \<inter> b) c P,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P whileAnno b {s. undefined} V c (P \<inter> - b),A"
qp_close_under_unitary_operator: fixes U :: "complex mat"   and dims :: "nat list"   and P :: "complex mat" assumes "U \<in> carrier_mat (state_sig.d dims) (state_sig.d dims)"   and "unitary U"   and "state_sig.is_quantum_predicate dims P" shows "state_sig.is_quantum_predicate dims (adjoint U * P * U)"
assms: fixes r' :: "C" assumes "r = C\<cdot>r'" shows "f\<cdot>r' = g\<cdot>r'"
wf_singleton: fixes a :: "'a"   and n :: "nat" assumes "a \<in> \<Sigma> n" shows "wf_atom n (singleton a)"
scale_zero_right: fixes a :: "'a" shows "scale a 0 = 0"
history: fixes ha :: "state \<Rightarrow> 'a"   and hb :: "'a \<Rightarrow> state \<times> state \<Rightarrow> 'a" shows "\<turnstile> \<exists>\<exists> h. Init h = ha \<and> \<box>(\<forall>x. $h = #x \<longrightarrow> h$ = hb x)"
thms_mono: fixes G :: "'b pl set"   and H :: "'b pl set" assumes "G \<subseteq> H" shows "(|-) G \<le> (|-) H"
TP_MST_def: fixes \<Sigma> :: "((Class \<times> Method) \<times> nat) list" shows "TP_MST \<Sigma> = (\<forall>C m. case MST\<down>(C, m) of None \<Rightarrow> \<Sigma>\<down>(C, m) = None | Some (T, MI, Anno) \<Rightarrow> Anno = emp \<and> (\<exists>n. \<Sigma>\<down>(C, m) = Some n \<and> (T, MI, Anno) = mkSPEC (Cachera n) emp))"
not_list_all_elim: fixes P :: "'a \<Rightarrow> bool"   and A :: "'a list" assumes "\<not> list_all P A" shows "\<exists>B x C. A = B @ x # C \<and> \<not> P x \<and> list_all P B"
C1_differentiable_on_def: fixes f :: "real \<Rightarrow> 'c"   and S :: "real set" shows "(f C1_differentiable_on S) = (\<exists>D. (\<forall>x\<in>S. (f has_vector_derivative D x) (at x)) \<and> continuous_on S D)"
affine_def: fixes s :: "'a set" shows "affine s = (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)"
rep_def: fixes r :: "'a" shows "rep r \<equiv> SOME \<phi>. \<guillemotleft>\<phi> : tab\<^sub>1 r \<star> (tab\<^sub>0 r)\<^sup>* \<Rightarrow> r\<guillemotright> \<and> local.iso \<phi>"
r_dettime_def: shows "r_dettime \<equiv> Mn 2 (Cn 3 r_and [Cn 3 r_less [recf.Id 3 2, Cn 3 r_length [Cn 3 r_pdec1 [Cn 3 r_state [recf.Id 3 0, recf.Id 3 1]]]], Cn 3 r_neq [recf.Id 3 2, Cn 3 r_pdec2 [Cn 3 r_state [recf.Id 3 0, recf.Id 3 1]]]])"
exec_cons: fixes s :: "'a"   and r :: "'b \<times> 'c"   and rs :: "('b \<times> 'c) list" shows "s \<star> Cons r rs = (s \<star> rs) \<bullet> r"
in_arcs_in_arcs: fixes x :: "'c"   and G :: "('d, 'c) pre_digraph"   and u :: "'d" assumes "x \<in> in_arcs G u" shows "x \<in> arcs G"
module_hom_fst: fixes s1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and s2 :: "'a \<Rightarrow> 'c \<Rightarrow> 'c" assumes "module_prod s1 s2" shows "module_hom (module_prod.scale s1 s2) s1 fst"
assms: shows "t \<in> trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (S \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>) \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>"
con1: fixes env :: "v sem_env"   and cn :: "(char list, char list) id0 option"   and es :: "exp0 list"   and rs :: "(v, v) result list"   and vs :: "v list"   and v0 :: "v" assumes "do_con_check (c env) cn (length es)"   and "list_all2_shortcircuit (cupcake_evaluate_single env) (rev es) rs"   and "sequence_result rs = Rval vs"   and "build_conv (c env) cn (rev vs) = Some v0" shows "cupcake_evaluate_single env (Con cn es) (Rval v0)"
ctxt_ctxt_compose: fixes C :: "('d, 'e) ctxt"   and D :: "('d, 'e) ctxt"   and t :: "('d, 'e) Term.term" shows "(C \<circ>\<^sub>c D)\<langle>t\<rangle> = C\<langle>D\<langle>t\<rangle>\<rangle>"
assign_eval\<^sub>w_load\<^sub>C: fixes x :: "var\<^sub>C"   and y :: "var\<^sub>C"   and mds :: "Mode \<Rightarrow> var\<^sub>C set"   and mem :: "var\<^sub>C \<Rightarrow> nat" shows "eval_abv\<^sub>C \<langle>x \<leftarrow> aexp\<^sub>C.Load y, mds, mem\<rangle>\<^sub>C \<langle>Stop, mds, mem(x := mem y)\<rangle>\<^sub>C"
Abs1: fixes x :: "'a"   and y :: "'b" assumes "T1 x y" shows "Abs1 x = y"
singleton_rev_conv: fixes x :: "'a"   and xs :: "'a list" shows "([x] = rev xs) = (xs = [x])"
intro: fixes \<Delta> :: "('a, 'b) network_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "flow_attainability \<Delta>"   and "flow_network_axioms \<Delta> g" shows "flow_network \<Delta> g"
embed_def: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set"   and f :: "'d \<Rightarrow> 'e" shows "embed r r' f \<equiv> \<forall>a\<in>Field r. bij_betw f (under r a) (under r' (f a))"
integrable_zero: fixes M :: "'b measure" shows "integrable M (\<lambda>x. 0::'c)"
Schottky_lemma1: fixes n :: "nat" assumes "0 < n" shows "0 < real n + sqrt ((real n)\<^sup>2 - 1)"
node\<^sub>i_cases: fixes xs :: "('b btree \<times> 'b) list"   and k :: "nat" shows "length xs \<le> k \<or> (\<exists>ls sub sep rs. BTree_Set.split_half xs = (ls, (sub, sep) # rs))"
a_comm: fixes x :: "rat vec"   and y :: "rat vec" assumes "x \<in> Rn"   and "y \<in> Rn" shows "x + y = y + x"
f_aggregate_append_mult: fixes xs :: "'d list"   and m :: "nat"   and k :: "nat"   and ys :: "'d list"   and ag :: "'d list \<Rightarrow> 'd" assumes "length xs = m * k" shows "f_aggregate (xs @ ys) k ag = f_aggregate xs k ag @ f_aggregate ys k ag"
path_from_root_Vr_ex: fixes v :: "'a" assumes "v \<in> verts G"   and "v \<noteq> s"   and "dist v \<noteq> \<infinity>" shows "\<exists>e. s \<rightarrow>\<^sup>* tail G e \<and> e \<in> arcs G \<and> head G e = v \<and> dist (tail G e) \<noteq> \<infinity> \<and> parent_edge v = Some e \<and> num v = num (tail G e) + 1"
row_space_eq_phull: fixes ps :: "('d \<Rightarrow>\<^sub>0 'e) list"   and ts :: "'d list" assumes "Keys (set ps) \<subseteq> set ts" shows "row_to_poly ts ` row_space (polys_to_mat ts ps) = phull (set ps)"
subgraph_in_degree: fixes H :: "('a, 'a \<times> 'a) pre_digraph"   and v :: "'a" assumes "subgraph H (with_proj G)" shows "in_degree H v \<le> in_degree (with_proj G) v"
pos_divide_le_eq: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and c :: "'d"   and b :: "'d"   and a :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero c" shows "less_eq (divide b c) a = less_eq b (times a c)"
Un_upper2: fixes B :: "'g set"   and A :: "'g set" shows "B \<subseteq> A \<union> B"
connected_space_discrete_topology: fixes U :: "'b set" shows "connected_space (discrete_topology U) = (\<exists>a. U \<subseteq> {a})"
simps_6: fixes typeof_addr :: "'a \<Rightarrow> 'b \<Rightarrow> htype option"   and P :: "'c prog"   and h :: "'a"   and v :: "'b val" shows "J1_heap_base.\<tau>move1 typeof_addr P h (Val v) = False"
Ndiv_def: shows "(\<div>\<^sub>N) = (\<lambda>a b. a *\<^sub>N Ninv b)"
section_retraction_compose: fixes e :: "'a"   and m :: "'a"   and e' :: "'a"   and m' :: "'a" assumes "ide (e \<cdot> m)"   and "ide (e' \<cdot> m')"   and "seq m' m" shows "ide ((e \<cdot> e') \<cdot> m' \<cdot> m)"
less_nat_zero_code: fixes n :: "nat" shows "((n::nat) < 0) = False"
count_filter_mset: fixes P :: "'a \<Rightarrow> bool"   and M :: "'a multiset"   and a :: "'a" shows "count (filter_mset P M) a = (if P a then count M a else 0)"
assms_3: shows "s \<in> dgrad_sig_set' (length fs) d"
dimension_def: shows "finite_dimensional_vector_space.dimension cart_basis = card cart_basis"
independent_explicit: fixes B :: "('b, 'c) vec set" shows "vec.independent B = (finite B \<and> (\<forall>c. (\<Sum>v\<in>B. c v *s v) = 0 \<longrightarrow> (\<forall>v\<in>B. c v = (0::'b))))"
simps_3: fixes t :: "'a E.term"   and u :: "'a E.term" shows "E.Arr (t \<^bold>\<star> u) = E.HSeq t u"
PiE_def: fixes S :: "'h set"   and T :: "'h \<Rightarrow> 'i set" shows "Pi\<^sub>E S T = Pi S T \<inter> extensional S"
disj: shows "A \<inter> U \<inter> (A \<inter> W - U) = {}"
next_in_basics_1: fixes A :: "'b set"   and a :: "'b"   and b :: "'b"   and u :: "'b" assumes "finite A"   and "A \<subseteq> {a..b}"   and "a \<in> A"   and "b \<in> A"   and "a < b"   and "u \<in> {a..<b}" shows "next_in A u \<in> A"
uniqueness_algorithm_division_to_int_poly: fixes g :: "int poly"   and F :: "'a mod_ring poly"   and f :: "int poly"   and G :: "'a mod_ring poly"   and Q :: "'a mod_ring poly"   and R :: "'a mod_ring poly"   and q :: "int poly"   and r :: "int poly" assumes "Mp g \<noteq> 0"   and "F = of_int_poly f"   and "G = of_int_poly g"   and "F = G * Q + R"   and "R = 0 \<or> degree R < degree G"   and "Mp f = Mp g * q + r"   and "r = 0 \<or> degree r < degree_m g" shows "Mp q = to_int_poly Q \<and> Mp r = to_int_poly R"
Tensor_preserves_coherent: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Arr t"   and "Arr u"   and "coherent t"   and "coherent u" shows "coherent (t \<^bold>\<otimes> u)"
simps_1: fixes uu :: "char list \<Rightarrow> 'c rule list option" shows "rewrite_Goto_chain_safe uu [] = Some []"
awalk_verts_conv': fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "pcas u p v" shows "pawalk_verts u p = (if p = [] then [u] else fst (hd p) # map snd p)"
lookup_inverse: fixes x :: "'b \<Rightarrow>\<^sub>0 'c" shows "Abs_poly_mapping (poly_mapping.lookup x) = x"
minus_add_minus_mat: fixes u :: "'a mat"   and nr :: "nat"   and nc :: "nat"   and v :: "'a mat"   and w :: "'a mat" assumes "u \<in> carrier_mat nr nc"   and "v \<in> carrier_mat nr nc"   and "w \<in> carrier_mat nr nc" shows "u - (v + w) = u - v - w"
adj_plus: fixes A :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and B :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a" shows "(A + B)* = A* + B*"
uint_mask_eq: fixes n :: "nat" shows "uint (mask n) = mask (min LENGTH('b) n)"
Arcsin_body_lemma: fixes z :: "complex" shows "\<i> * z + csqrt (1 - z\<^sup>2) \<noteq> 0"
simps_2: fixes c :: "'form"   and uu :: "nat"   and "is" :: "nat list" shows "vidx (c, uu # is) = iAnnot (tree_at (it' c) is)"
ys: shows "ys \<equiv> as @ pair_relator_halflist t s @ bs"
lm04: fixes P :: "'c \<Rightarrow> bool"   and l :: "'c list" shows "concat (map (\<lambda>n. if P n then [n] else []) l) = concat (map (\<lambda>n. if n \<in> set l then if P n then [n] else [] else []) l)"
simps_2: fixes n :: "nat"   and bs :: "bool list"   and bss :: "bool list list" shows "nats_of_boolss n (bs # bss) = map2 (\<lambda>b x. of_bool b + 2 * x) bs (nats_of_boolss n bss)"
maxLemma: fixes x :: "'c"   and X :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "x \<in> X"   and "finite X" shows "f x \<le> Max (f ` X)"
sublist_filter: fixes P :: "'a \<Rightarrow> bool"   and l :: "'a list" shows "subseq (filter P l) l"
Confidentiality_Serv_A: fixes A :: "agent"   and authK :: "nat"   and Ta :: "nat"   and authTicket :: "msg"   and evs :: "event list"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg" assumes "Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta, authTicket\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace> \<in> parts (knows Spy evs)"   and "\<not> expiredAK Ta evs"   and "\<not> expiredSK Ts evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> kerbIV" shows "Key servK \<notin> analz (knows Spy evs)"
showsp_complex_def: fixes p :: "nat"   and x :: "complex"   and y :: "char list" shows "showsp_complex p x y = show_complex x @ y"
order_prop: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (\<exists>z. x + z = y)"
simps_1: fixes uu :: "nat" shows "flip uu [] = []"
rep_inverse: fixes x :: "('b, 'c) mapping" shows "Mapping (mapping.rep x) = x"
simps_54: fixes f1 :: "bool \<Rightarrow> 'a"   and f2 :: "int \<Rightarrow> 'a"   and f3 :: "int \<Rightarrow> 'a"   and f4 :: "int \<Rightarrow> 'a"   and f5 :: "CTypeId \<Rightarrow> nat \<Rightarrow> 'a"   and f6 :: "Arraytype \<Rightarrow> nat \<Rightarrow> 'a"   and f7 :: "'a"   and x61 :: "Arraytype"   and x62 :: "nat" shows "(case arrV x61 x62 of boolV x \<Rightarrow> f1 x | intgV x \<Rightarrow> f2 x | shortV x \<Rightarrow> f3 x | byteV x \<Rightarrow> f4 x | objV x xa \<Rightarrow> f5 x xa | arrV x xa \<Rightarrow> f6 x xa | nullV \<Rightarrow> f7) = f6 x61 x62"
run_put_option: fixes put :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and s :: "'c"   and m :: "('b, 'a) optionT" shows "run_option (put_option put s m) = put s (run_option m)"
simps: fixes i :: "int"   and j :: "int" shows "List.upto i j = (if i \<le> j then i # List.upto (i + 1) j else List.list.Nil)"
NonceCCA_secrecy_lemma: fixes i :: "nat"   and evs :: "event list"   and K :: "nat"   and C :: "agent"   and N :: "nat"   and NonceCCA :: "nat"   and X :: "msg"   and Y :: "msg" assumes "CA i \<notin> bad"   and "evs \<in> set_cr"   and "Key K \<notin> analz (knows Spy evs)"   and "Crypt K \<lbrace>sign (priSK (CA i)) \<lbrace>Agent C, Nonce N, Agent (CA i), Nonce NonceCCA\<rbrace>, X, Y\<rbrace> \<in> parts (knows Spy evs)" shows "Nonce NonceCCA \<notin> analz (knows Spy evs)"
rel_stacktraces_Nil: fixes get :: "'c \<Rightarrow> ('d, 'e, 'c, 'op, 'opinl, 'opubx, 'f, 'g) Ubx.instr fundef option"   and opt :: "'c option" shows "rel_stacktraces get [] [] opt"
par_comp\<^sub>l\<^sub>s\<^sub>s\<^sub>t_proj: fixes A :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list"   and Sec :: "('fun, 'var) Term.term set"   and n :: "'lbl" assumes "par_comp\<^sub>l\<^sub>s\<^sub>s\<^sub>t A Sec" shows "par_comp\<^sub>l\<^sub>s\<^sub>s\<^sub>t (proj n A) Sec"
face_cycle_pred_def: shows "face_cycle_pred \<equiv> edge_rev M \<circ> edge_pred M"
no_real_normal_subgroup: fixes H :: "'a set" assumes "H \<lhd> G" shows "H = carrier G \<or> H = {\<one>}"
continuous_map_compose: fixes X :: "'b topology"   and X' :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and X'' :: "'d topology"   and g :: "'c \<Rightarrow> 'd" assumes "continuous_map X X' f"   and "continuous_map X' X'' g" shows "continuous_map X X'' (g \<circ> f)"