hcju/isabelleps huggingface dataset:
Zp_residue_eq: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and k :: "nat" assumes "a \<in> carrier Zp"   and "b \<in> carrier Zp"   and "eint (int k) < val_Zp (a \<ominus> b)" shows "a k = b k"
small_Union_iff: fixes X :: "V set" shows "small (\<Union> (elts ` X)) = small X"
Infinitesimal_hrealpow: fixes x :: "real star"   and n :: "nat" assumes "x \<in> Infinitesimal"   and "0 < n" shows "x ^ n \<in> Infinitesimal"
dom_map_update: fixes m :: "'c \<Rightarrow> 'd option"   and k :: "'c"   and v :: "'d" shows "dom (m(k \<mapsto> v)) = dom m \<union> {k}"
pred_inject_1: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a" shows "pred_M1 P (M1.Empty a) = P a"
onorm_le_matrix_component_sum: fixes A :: "((real, 'a) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec" shows "onorm ((*v) A) \<le> (\<Sum>i\<in>UNIV. \<Sum>j\<in>UNIV. \<bar>A $h i $h j\<bar>)"
f: shows "\<forall>x. eval f [x] \<down>= (if x \<in> {x. eval (code x) [x] \<down>} then 1 else 0)"
filter_eq_iff: fixes F :: "'c filter"   and F' :: "'c filter" shows "(F = F') = (\<forall>P. eventually P F = eventually P F')"
psimps_2: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and p :: "('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat"   and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list"   and z :: "nat" assumes "rb_aux_dom ((bs, ss, p # ps), z)" shows "rb_aux ((bs, ss, p # ps), z) = (let ss' = new_syz_sigs ss bs p in if sig_crit bs ss' p then rb_aux ((bs, ss', ps), z) else let p' = sig_trd bs (poly_of_pair p) in if rep_list p' = 0 then rb_aux ((bs, lt p' # ss', ps), Suc z) else rb_aux ((p' # bs, ss', add_spairs ps bs p'), z))"
maxne0_def: fixes x :: "nat list"   and a :: "nat list" shows "maxne0 x a = (if length x = length a \<and> (\<exists>i<length a. x ! i \<noteq> 0) then Max {a ! i |i. i < length a \<and> x ! i \<noteq> 0} else 0)"
i_def: shows "i \<equiv> \<l>[\<I>\<^sub>1] \<cdot> C\<^sub>1.runit' \<I>"
r: shows "r = \<langle>rl, (ra, rlv), rr\<rangle>"
ind: assumes "abc_notfinal (abc_steps_l (0, lm) A n) A" shows "abc_steps_l (0, lm) (A [+] B) n = abc_steps_l (0, lm) A n"
Abs_zet_inverse: fixes y :: "'a set" assumes "y \<in> zet" shows "Rep_zet (Abs_zet y) = y"
coshr_def: fixes x :: "real" shows "coshr x \<equiv> st (sumhr (0, whn, \<lambda>n. cos_coeff n * x ^ n))"
gap_precond_def: fixes l :: "nat \<Rightarrow> 'b"   and k :: "'b" shows "gap_precond l k \<equiv> \<forall>v\<in>V. l v \<noteq> k"
usolves_odeI: fixes y :: "real \<Rightarrow> 'c"   and f :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "real set"   and X :: "'c set"   and t0 :: "real" assumes "(y solves_ode f) T X"   and "t0 \<in> T"   and "is_interval T"   and "\<And>z t. \<lbrakk>{t0--t} \<subseteq> T; (z solves_ode f) {t0--t} X; z t0 = y t0\<rbrakk> \<Longrightarrow> z t = y t" shows "(y usolves_ode f from t0) T X"
r2f_Sup_pres_var: fixes R :: "('a \<times> 'b) set set" shows "\<F> (\<mu> R) = \<Squnion> \<P> \<F> R"
trms_list\<^sub>s\<^sub>s\<^sub>t_def: fixes S :: "('c, 'd) stateful_strand_step list" shows "trms_list\<^sub>s\<^sub>s\<^sub>t S \<equiv> remdups (concat (map trms_list\<^sub>s\<^sub>s\<^sub>t\<^sub>p S))"
SMP_SMP_subset: fixes N :: "('fun, 'var) Term.term set"   and M :: "('fun, 'var) Term.term set" assumes "N \<subseteq> SMP M" shows "SMP N \<subseteq> SMP M"
r: fixes i :: "'a" assumes "i \<in> K" shows "0 < r i \<and> r i \<le> B"
FOREACHoi_def: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and \<Phi> :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and S :: "'c set" shows "FOREACH\<^sub>O\<^bsup>R,\<Phi>\<^esup> S \<equiv> FOREACH\<^sub>O\<^sub>C\<^bsup>R,\<Phi>\<^esup> S (\<lambda>_. True)"
smodule_ideal_coeff_is_Submodule: fixes A :: "'b set" assumes "ideal R A" shows "submodule R M (A \<odot>\<^bsub>R\<^esub> M)"
simps_4: fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and C :: "('f, 'd, 'e) psiCase" assumes "assertionAux SCompose SImp SBottom SChanEq" shows "assertionAux.extractFrame SCompose SBottom (Case C) = \<langle>\<epsilon>, SBottom\<rangle>"
closed_lift2_sup: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b err"   and n :: "nat" assumes "closed (err A) (lift2 f)" shows "closed (err (list n A)) (lift2 (Listn.sup f))"
qGoodAbs_iff_goodAbs_asAbs: fixes qA :: "('d, 'e, 'varSort, 'var, 'f) qAbs" shows "goodAbs (asAbs qA) = qGoodAbs qA"
ladder_n_simp1: fixes u :: "nat"   and L :: "(nat \<times> nat \<times> nat) list"   and L' :: "(nat \<times> nat \<times> nat) list" assumes "u < length L" shows "ladder_n (L @ L') u = ladder_n L u"
closed: fixes A :: "ordinal set"   and g :: "nat \<Rightarrow> ordinal" assumes "normal_set A"   and "\<forall>n. g n \<in> A" shows "oLimit g \<in> A"
induct: fixes P :: "e list \<Rightarrow> bool"   and a0 :: "e list" assumes "\<And>v es. P es \<Longrightarrow> P (($C v) # es)"   and "P []"   and "\<And>va. P (($Unreachable) # va)"   and "\<And>va. P (($Nop) # va)"   and "\<And>va. P (($Drop) # va)"   and "\<And>va. P (($Select) # va)"   and "\<And>vc vd va. P (($Block vc vd) # va)"   and "\<And>vc vd va. P (($Loop vc vd) # va)"   and "\<And>vc vd ve va. P (($b_e.If vc vd ve) # va)"   and "\<And>vc va. P (($Br vc) # va)"   and "\<And>vc va. P (($Br_if vc) # va)"   and "\<And>vc vd va. P (($Br_table vc vd) # va)"   and "\<And>va. P (($Return) # va)"   and "\<And>vc va. P (($Call vc) # va)"   and "\<And>vc va. P (($Call_indirect vc) # va)"   and "\<And>vc va. P (($Get_local vc) # va)"   and "\<And>vc va. P (($Set_local vc) # va)"   and "\<And>vc va. P (($Tee_local vc) # va)"   and "\<And>vc va. P (($Get_global vc) # va)"   and "\<And>vc va. P (($Set_global vc) # va)"   and "\<And>vc vd ve vf va. P (($Load vc vd ve vf) # va)"   and "\<And>vc vd ve vf va. P (($Store vc vd ve vf) # va)"   and "\<And>va. P (($Current_memory) # va)"   and "\<And>va. P (($Grow_memory) # va)"   and "\<And>vc vd va. P (($Unop_i vc vd) # va)"   and "\<And>vc vd va. P (($Unop_f vc vd) # va)"   and "\<And>vc vd va. P (($Binop_i vc vd) # va)"   and "\<And>vc vd va. P (($Binop_f vc vd) # va)"   and "\<And>vc vd va. P (($Testop vc vd) # va)"   and "\<And>vc vd va. P (($Relop_i vc vd) # va)"   and "\<And>vc vd va. P (($Relop_f vc vd) # va)"   and "\<And>vc vd ve vf va. P (($Cvtop vc vd ve vf) # va)"   and "\<And>va. P (Trap # va)"   and "\<And>vb va. P (Callcl vb # va)"   and "\<And>vb vc vd va. P (Label vb vc vd # va)"   and "\<And>vb vc vd ve va. P (Local vb vc vd ve # va)" shows "P a0"
decode_Var_q_Var: fixes i :: "name" shows "decode_Var (q_Var i) = i"
state_rank_Some: fixes q :: "'b"   and n :: "nat" assumes "\<not> sink q" shows "(\<exists>i. state_rank q n = Some i) = (\<exists>j. oldest_token q n = Some j)"
rel_inject_1: fixes R1 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and R2 :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and R3 :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and x1 :: "'g"   and y1 :: "'h" shows "rel_generat R1 R2 R3 (Pure x1) (Pure y1) = R1 x1 y1"
nth_mem_mset: fixes i :: "nat"   and ls :: "'c list" assumes "i < length ls" shows "ls ! i \<in># mset ls"
env_C_restr_cong: fixes r :: "C"   and f :: "'a \<Rightarrow> C \<rightarrow> 'b"   and g :: "'a \<Rightarrow> C \<rightarrow> 'b" assumes "\<And>x r'. r' \<sqsubseteq> r \<Longrightarrow> f x\<cdot>r' = g x\<cdot>r'" shows "f|\<^sup>\<circ>\<^bsub>r\<^esub> = g|\<^sup>\<circ>\<^bsub>r\<^esub>"
meet_all_3: fixes l :: "Lines"   and m :: "Lines"   and n :: "Lines"   and P :: "Points" assumes "meet_3_in l m n P" shows "meet_in m n P"
union: fixes A :: "'b set"   and B :: "'b set" assumes "finite A"   and "A \<noteq> {}"   and "finite B"   and "B \<noteq> {}" shows "Max (A \<union> B) = max (Max A) (Max B)"
jvm_refl: fixes P :: "jvm_prog"   and \<sigma> :: "jvm_state" shows "P \<turnstile> \<sigma> -jvm\<rightarrow> \<sigma>"
moebius_pt_moebius_id: shows "moebius_pt id_moebius = id"
processedD3: fixes a :: "'e"   and b :: "'e"   and xs :: "'e list"   and ps :: "('e \<times> 'e) list" assumes "processed (a, b) xs ps" shows "(a, b) \<notin>\<^sub>p set ps"
chain1_eq_chain2: shows "chain_reparam_chain' one_chain1 subdiv"
stable_rank_equiv: fixes x :: "nat" shows "(\<exists>i. stable_rank x i) = (\<exists>n. stable_rank_at x n)"
sylvester_mat_sub_map: fixes f :: "'d \<Rightarrow> 'c"   and m :: "nat"   and n :: "nat"   and p :: "'d poly"   and q :: "'d poly" assumes "f (0::'d) = (0::'c)" shows "map_mat f (sylvester_mat_sub m n p q) = sylvester_mat_sub m n (map_poly f p) (map_poly f q)"
catch_gpv_map': fixes f :: "'g \<Rightarrow> 'd"   and g :: "'h \<Rightarrow> 'e"   and h :: "'f \<Rightarrow> 'i"   and gpv :: "('g, 'h, 'i) gpv" shows "catch_gpv (map_gpv' f g h gpv) = map_gpv' (map_option f) g h (catch_gpv gpv)"
intro: fixes r :: "('c \<times> 'c) set" assumes "Well_order r" shows "wo_rel r"
dependent_mono: fixes A :: "'b set"   and B :: "'b set" assumes "local.dependent A"   and "A \<subseteq> B" shows "local.dependent B"
genby_order2': fixes s :: "'a"   and w :: "'a" assumes "s + s = (0::'a)"   and "w \<in> \<langle>{s}\<rangle>" shows "w = (0::'a) \<or> w = s"
agree_on_comm: fixes X :: "('b, 'c) labeled_graph"   and f :: "('c \<times> 'd) set"   and g :: "('c \<times> 'd) set" shows "agree_on X f g = agree_on X g f"
SIod_sub_sub: fixes S :: "'c set"   and T :: "'c set"   and D :: "'c Order" assumes "S \<subseteq> T"   and "T \<subseteq> carrier D" shows "SIod (SIod D T) S = SIod D S"
CatIdDomCod_2: fixes C :: "('a, 'b, 'c) Category_scheme"   and X :: "'a" assumes "Category C"   and "X \<in> obj\<^bsub>C\<^esub>" shows "cod\<^bsub>C\<^esub> (id\<^bsub>C\<^esub> X) = X"
cr_integer_def: shows "cr_integer \<equiv> \<lambda>x y. x = int_of_integer y"
less_measure_def: fixes M :: "'a measure"   and N :: "'a measure" shows "(M < N) = (M \<le> N \<and> \<not> N \<le> M)"
stutter_equiv_within_interval: fixes f :: "nat \<Rightarrow> nat"   and \<sigma> :: "nat \<Rightarrow> 'b"   and k :: "nat"   and n :: "nat" assumes "stutter_sampler f \<sigma>"   and "f k \<le> n"   and "n < f (Suc k)" shows "\<sigma> [n..] \<approx> \<sigma> [f k..]"
in_set_root: fixes x :: "'c"   and l :: "'c tree"   and r :: "'c tree" shows "x \<in> set_of (Node l x False r)"
hom_one: fixes h :: "'e \<Rightarrow> 'f"   and G :: "('e, 'g) monoid_scheme"   and H :: "('f, 'h) monoid_scheme" assumes "h \<in> hom G H"   and "Group.group G"   and "Group.group H" shows "h \<one>\<^bsub>G\<^esub> = \<one>\<^bsub>H\<^esub>"
lfilter_empty_conv: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a llist" shows "(lfilter P xs = []\<^sub>l) = (\<forall>x\<in>lset xs. \<not> P x)"
flat_snth: fixes s :: "'d list stream"   and n :: "nat" assumes "\<forall>xs\<in>sset s. xs \<noteq> []" shows "flat s !! n = (if n < length (shd s) then shd s ! n else flat (stl s) !! (n - length (shd s)))"
antisym: fixes a :: "'a"   and b :: "'a" assumes "a \<preceq> b"   and "b \<preceq> a" shows "a = b"
\<nu>_in_hom_2: shows "\<guillemotleft>\<nu> : local.dom \<omega> \<Rightarrow> g \<star> w\<guillemotright>"
tllist_ord_TCons_TCons: fixes x :: "'c"   and xs :: "('c, 'b) tllist"   and y :: "'c"   and ys :: "('c, 'b) tllist" shows "local.tllist_ord (TCons x xs) (TCons y ys) = (x = y \<and> local.tllist_ord xs ys)"
ivl_disj_int_4: fixes l :: "'c"   and u :: "'c" shows "{..<l} \<inter> {l..u} = {}"
axioms_2: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "pseudo_ordered_set A less_eq" shows "antisymmetric A less_eq"
o_assoc: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'd"   and h :: "'b \<Rightarrow> 'e" shows "f \<circ> (g \<circ> h) = f \<circ> g \<circ> h"
admissibleI: fixes ord :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and P :: "'f \<Rightarrow> bool"   and lub :: "'f set \<Rightarrow> 'f" assumes "\<And>A. \<lbrakk>Complete_Partial_Order.chain ord A; A \<noteq> {}; \<forall>x\<in>A. P x\<rbrakk> \<Longrightarrow> P (lub A)" shows "ccpo.admissible lub ord P"
sum_closed: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'a" assumes "\<And>x. x \<in> X \<Longrightarrow> f x \<in> A" shows "sum f X \<in> A"
WT_callee: fixes callee :: "'d \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'd) spmf"   and I :: "'d \<Rightarrow> bool"   and \<I> :: "('e, 'f) \<I>"   and s :: "'d" assumes "callee_invariant_on callee I \<I>"   and "I s" shows "\<I> \<turnstile>c callee s \<surd>"
monotoneE: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "Tarski.monotone f A r"   and "x \<in> A"   and "y \<in> A"   and "(x, y) \<in> r" shows "(f x, f y) \<in> r"
cfg_of_cfg_onI: fixes K :: "'a \<Rightarrow> 'a pmf set"   and D :: "'a pmf"   and s :: "'a"   and c :: "'a \<Rightarrow> 'a cfg" assumes "Markov_Decision_Process K"   and "D \<in> K s"   and "\<And>t. t \<in> set_pmf D \<Longrightarrow> c t \<in> Markov_Decision_Process.cfg_on K t" shows "Cfg s D c \<in> Markov_Decision_Process.cfg_on K s"
lambda_predicates_2_3: fixes \<phi> :: "\<kappa> \<Rightarrow> \<kappa> \<Rightarrow> \<kappa> \<Rightarrow> \<o>"   and x :: "\<nu>"   and y :: "\<nu>"   and z :: "\<nu>" assumes "IsProperInXYZ \<phi>" shows "[[\<lparr>\<^bold>\<lambda>\<^sup>3 (\<lambda>x y z. \<phi> (x\<^sup>P) (y\<^sup>P) (z\<^sup>P)),x\<^sup>P,y\<^sup>P,z\<^sup>P\<rparr> \<^bold>\<equiv> \<phi> (x\<^sup>P) (y\<^sup>P) (z\<^sup>P)]]"
naturality: fixes f :: "'a" assumes "C.arr f" shows "C.map (C.cod f) \<cdot> C.map f = C.map f \<cdot> C.map (C.dom f)"
rplusinf_uset: fixes p :: "fm"   and a :: "real"   and bs :: "real list"   and x :: "real" assumes "isrlfm p"   and "\<not> Ifm (a # bs) (plusinf p)"   and "Ifm (x # bs) p" shows "\<exists>(s, m)\<in>set (uset p). x \<le> Inum (a # bs) s / real_of_int m"
prems_3: shows "s1 = t1 (\<le> l)"
mult_oner: fixes x :: "'a" shows "x \<parallel> 1\<^sub>\<pi> = x"
comp: fixes f :: "'k \<Rightarrow> 'i"   and g :: "'l \<Rightarrow> 'j"   and h :: "'g \<Rightarrow> 'k"   and i :: "'h \<Rightarrow> 'l" shows "map_prod f g \<circ> map_prod h i = map_prod (f \<circ> h) (g \<circ> i)"
Ioi_le_Ico: fixes a :: "'a multiset" shows "subset_mset.greaterThan a \<subseteq> subset_mset.atLeast a"
cone_axioms: shows "\<chi>a.D.cone (l a) (\<chi> a)"
bind_split: fixes P :: "'d option \<Rightarrow> bool"   and m :: "'e option"   and f :: "'e \<Rightarrow> 'd option" shows "P (Option.bind m f) = ((m = None \<longrightarrow> P None) \<and> (\<forall>v. m = Some v \<longrightarrow> P (f v)))"
lookup_tabulate: fixes x :: "'c"   and xs :: "'c list"   and f :: "'c \<Rightarrow> 'd" assumes "x \<in> set xs" shows "Mapping.lookup (Mapping.tabulate xs f) x = Some (f x)"
image_Int: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'b set" assumes "inj f" shows "f ` (A \<inter> B) = f ` A \<inter> f ` B"
feq: shows "range (flow0 x) = flow0 x ` {0..T}"
arden_l_counter: shows "\<exists>x y z. x = y \<cdot> x + z \<and> x \<noteq> y\<^sup>\<star> \<cdot> z"
true_disj_zero_1: fixes P :: "'a upred" shows "(P \<or> true) = true"
finite_lprefix_nitpick_simps_2: fixes xs :: "'c llist" shows "Coinductive_List.finite_lprefix LNil xs = True"
1: shows "Var xs x #[rho] = asTerm (case pickE rho xs x of None \<Rightarrow> qVar xs x | Some qY \<Rightarrow> qY)"
index_mat_swapcols_3: fixes k :: "nat"   and l :: "nat"   and A :: "'a mat" shows "dim_col (swapcols k l A) = dim_col A"
mat_cons4: fixes p :: "pat"   and ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and v1 :: "v"   and e :: "exp0"   and pes :: "(pat \<times> exp0) list"   and err_v :: "v" assumes "\<not> allDistinct (pat_bindings p [])" shows "evaluate_match ck env s v1 ((p, e) # pes) err_v (s, Rerr (Rabort Rtype_error))"
inj_aux: fixes x :: "'a" assumes "x \<in> carrier G"   and "double_dual_iso x = \<one>\<^bsub>Characters (Characters G)\<^esub>" shows "x = \<one>"
group_inv_rel: fixes G :: "('c, 'd) monoid_scheme"   and \<phi> :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and M :: "'e set"   and g :: "'c"   and m :: "'e"   and n :: "'e" assumes "group_action G \<phi> M"   and "g \<in> carrier G"   and "m \<in> M"   and "n \<in> M"   and "\<phi> g n = m" shows "\<phi> (inv\<^bsub>G\<^esub> g) m = n"
is_functor: shows "functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) C.map"
assms: shows "y \<cdot> x \<le> x \<cdot> (x + y)\<^sup>\<star>"
max_x'_le_drop: fixes y :: "nat list"   and b :: "nat list"   and a :: "nat list"   and i :: "nat"   and l :: "nat" assumes "length y \<le> length b" shows "hlde_ops.max_x' a b y i \<le> hlde_ops.max_x' a b (drop l y) i"
conf_upd_obj: fixes h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and a :: "nat"   and C :: "char list"   and S :: "(char list list \<times> (char list \<Rightarrow> val option)) set"   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"   and S' :: "(char list list \<times> (char list \<Rightarrow> val option)) set"   and v :: "val"   and T :: "ty" assumes "h a = \<lfloor>(C, S)\<rfloor>" shows "(P,h(a \<mapsto> (C, S')) \<turnstile> v :\<le> T) = (P,h \<turnstile> v :\<le> T)"
reachable_def: fixes x :: "(recf \<times> nat list \<times> nat list) list \<times> nat option"   and y :: "(recf \<times> nat list \<times> nat list) list \<times> nat option" shows "reachable x y \<equiv> \<exists>t. iterate t step x = y"
dom_eq_dom_share_eq: fixes S :: "nat \<Rightarrow> bool option"   and S' :: "nat \<Rightarrow> bool option"   and sb :: "'b memref list" assumes "dom S = dom S'" shows "dom (share sb S) = dom (share sb S')"
normal_vector_5: fixes W :: "'a vec set" assumes "finite W"   and "Suc (card W) = n"   and "lin_indpt W"   and "W \<subseteq> carrier_vec n" shows "lin_indpt (insert (normal_vector W) W)"
mem_Collect_eq: fixes a :: "'h"   and P :: "'h \<Rightarrow> bool" shows "(a \<in> Collect P) = P a"
wb_lens_weak: fixes x :: "'d \<Longrightarrow> 'e" assumes "wb_lens x" shows "weak_lens x"
simps_6: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> 'a"   and x2 :: "'c" shows "(case Inr x2 of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = f2 x2"
span_clauses_1: fixes a :: "'a"   and S :: "'a set" assumes "a \<in> S" shows "a \<in> ideal S"
FL_valid_Tree\<^sub>\<alpha>_Act\<^sub>\<alpha>: fixes P :: "'state"   and f :: "'effect"   and \<alpha> :: "'act"   and t\<^sub>\<alpha> :: "('b, 'pred, 'act, 'effect) Tree\<^sub>\<alpha>" shows "FL_valid_Tree\<^sub>\<alpha> P (Act\<^sub>\<alpha> f \<alpha> t\<^sub>\<alpha>) = (\<exists>\<alpha>'' t\<^sub>\<alpha>' P'. Act\<^sub>\<alpha> f \<alpha> t\<^sub>\<alpha> = Act\<^sub>\<alpha> f \<alpha>'' t\<^sub>\<alpha>' \<and> \<langle>f\<rangle>P \<rightarrow> \<langle>\<alpha>'',P'\<rangle> \<and> FL_valid_Tree\<^sub>\<alpha> P' t\<^sub>\<alpha>')"
maximal_independent_subset: fixes V :: "('b, 'c) vec set" obtains B :: "('b, 'c) vec set" where "B \<subseteq> V" and "vec.independent B" and "V \<subseteq> vec.span B"
atI_def: fixes a :: "char list"   and G :: "igraph"   and l :: "location" shows "a @\<^bsub>G\<^esub> l \<equiv> a \<in> agra G l"
get_local: fixes i :: "nat"   and \<C> :: "t_context"   and t :: "t" assumes "i < length (local \<C>)"   and "local \<C> ! i = t" shows "\<C> \<turnstile> [Get_local i] : [] _> [t]"
Joints_aforms_of_ivlsD1: fixes zs :: "real list"   and xs :: "real list"   and ys :: "real list" assumes "zs \<in> Joints (aforms_of_ivls xs ys)"   and "list_all2 (\<le>) xs ys" shows "list_all2 (\<le>) xs zs"
n_distrib1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "n x \<cdot> (n y \<oplus> n z) = n x \<cdot> n y \<oplus> n x \<cdot> n z"
plus_Box_def: fixes x :: "'a Box"   and y :: "'a Box" shows "x + y \<equiv> bliftM2 (\<Lambda> a b. a + b)\<cdot>x\<cdot>y"
mon_e_simps_4: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and p :: "'c" shows "mon_e fg (LSpawn p) = {}"
all_conj_distrib: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "(\<forall>x. P x \<and> Q x) = ((\<forall>x. P x) \<and> (\<forall>x. Q x))"
linorder_cases: fixes x :: "'a Word.word"   and y :: "'a Word.word" obtains "x <s y"   | "x = y"   | "y <s x"
vpa_snoc_Call: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r :: "'e"   and p :: "'f"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list"   and cs :: "'a list"   and as :: "'a list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "CFG.valid_path_aux kind get_return_edges cs as" shows "CFG.valid_path_aux kind get_return_edges cs (as @ [a])"
filterACS: shows "set (filter_compliant_stateful_ACS G M edgesList) = edges G - backflows (edges G)"
MPoly_induct: fixes P :: "'b mpoly \<Rightarrow> bool"   and x :: "'b mpoly" assumes "\<And>y. y \<in> UNIV \<Longrightarrow> P (MPoly y)" shows "P x"
2: fixes x :: "'b \<times> 'c"   and xa :: "'b"   and y :: "'c" assumes "x = f kv"   and "(xa, y) = x" shows "set (map_pair f xs) \<subseteq> f ` set xs"
sin_kpi: fixes k :: "int" shows "sin (real_of_int k * pi) = 0"
vmc_path_ltl: shows "vmc_path G (ltl P) w0 p \<sigma>"
simps_2: fixes n :: "nat" shows "Common.enat_of_option (Some n) = enat n"
dnso4: fixes x :: "'a"   and y :: "'a" shows "ans_d (x + y) = ans_d x + ans_d y"
assms_1: shows "length (labels \<sigma>) \<le> 1"
interpret_floatarith_Nr: fixes U :: "real"   and vs :: "real list" shows "interpret_floatarith (N\<^sub>r U) vs = real_of_float (float_of U)"
h: shows "((\<Sum>i = 0..<d + 1. f i / C ^ i) = 0) = ((\<Sum>i = 0..<d + 1. f i * C ^ (d - i)) = 0)"
matrix_mul_lid: fixes A :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec" shows "mat (1::'d) ** A = A"
permutes_imp_bij': fixes p :: "'a \<Rightarrow> 'a"   and S :: "'a set" assumes "p permutes S" shows "bij p"
rel_conversep: fixes R1 :: "'i \<Rightarrow> 'f \<Rightarrow> bool"   and R2 :: "'j \<Rightarrow> 'g \<Rightarrow> bool" shows "Resumption.resumption.rel_resumption R1\<inverse>\<inverse> R2\<inverse>\<inverse> = (Resumption.resumption.rel_resumption R1 R2)\<inverse>\<inverse>"
res_mult_eq: fixes x :: "int"   and y :: "int" shows "x \<otimes> y = x * y mod m"
filter_distr_rev: fixes f :: "'a \<Rightarrow> bool"   and lst :: "'a list" shows "filter f (rev lst) = rev (filter f lst)"
IH: assumes "A \<in> set F"   and "form_of_lit A \<^bold>\<or> disj_of_clause (removeAll A F), \<Gamma> \<Rightarrow>\<^sub>n" shows "disj_of_clause F, \<Gamma> \<Rightarrow>\<^sub>n"
prefix_augment: fixes p :: "'c list"   and n :: "nat"   and s :: "'c list \<Rightarrow> 'c" shows "prefix p ((augment_list s ^^ n) p)"
distinct_1: fixes \<Delta> :: "('a, 'b) network_scheme"   and x1 :: "'c"   and x21 :: "'c"   and x22 :: "'c" assumes "antiparallel_edges \<Delta>" shows "antiparallel_edges.Vertex x1 \<noteq> antiparallel_edges.Edge x21 x22"