hcju/isabelleps huggingface dataset:
orderI: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "Semilat A r f" shows "order r"
2: fixes xa :: "'a list"   and y :: "'a list" assumes "(xa, y) = partition (\<lambda>y. f y = f x) xs" shows "distinct (map f (local.group y))"
eq: shows "ln (fact (2 * n)) - 2 * ln (fact n) = prime_sum_upto (\<lambda>p. real_of_int (S n p) * ln (real p)) (real (2 * n))"
next_Times: fixes fsm1 :: "('b, 'c) fsm_scheme"   and fsm2 :: "('b, 'd) fsm_scheme"   and st1 :: "hf"   and st2 :: "hf"   and x :: "'b"   and st' :: "hf" shows "next (Times fsm1 fsm2) \<langle>st1, st2\<rangle> x st' = (\<exists>st1' st2'. st' = \<langle>st1', st2'\<rangle> \<and> next fsm1 st1 x st1' \<and> next fsm2 st2 x st2')"
prod_topology_subtopology_eu: fixes S :: "'b set"   and T :: "'c set" shows "prod_topology (top_of_set S) (top_of_set T) = top_of_set (S \<times> T)"
polish_pmpt_def: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" shows "polish_pmpt M T \<equiv> pmpt M T \<and> polish_pmpt_axioms M"
mem_set_list_relE2: fixes xs :: "'d list"   and ys :: "'e list"   and R :: "('d \<times> 'e) set"   and y :: "'e" assumes "(xs, ys) \<in> \<langle>R\<rangle>list_rel"   and "y \<in> set ys" obtains x :: "'d" where "x \<in> set xs" and "(x, y) \<in> R"
of_nat_lt_\<omega>: fixes n :: "nat" shows "of_nat n < \<omega>"
disjoint_iff_not_equal: fixes A :: "'d set"   and B :: "'d set" shows "(A \<inter> B = {}) = (\<forall>x\<in>A. \<forall>y\<in>B. x \<noteq> y)"
br_def: fixes \<alpha> :: "'e \<Rightarrow> 'f"   and I :: "'e \<Rightarrow> bool" shows "br \<alpha> I \<equiv> {(c, a). a = \<alpha> c \<and> I c}"
analz_synth_Un2: fixes G :: "msg set"   and H :: "msg set" shows "analz (G \<union> synth H) = analz (G \<union> H) \<union> synth H"
vbrelation_Limit_in_VsetI: fixes \<alpha> :: "V" assumes "Limit \<alpha>"   and "\<D>\<^sub>\<circ> (\<CC>\<lparr>Dom\<rparr>) \<in>\<^sub>\<circ> Vset \<alpha>"   and "\<R>\<^sub>\<circ> (\<CC>\<lparr>Dom\<rparr>) \<in>\<^sub>\<circ> Vset \<alpha>" shows "\<CC>\<lparr>Dom\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
no_imp_one: fixes x :: "'b"   and p :: "'b list" assumes "no_call_in_path x p" shows "one_call_in_path x p"
deriv: fixes t :: "real" assumes "0 < t" shows "(f has_real_derivative f' t) (at t)"
interior_of_eq: fixes X :: "'a topology"   and S :: "'a set" shows "(X interior_of S = S) = openin X S"
cdf_lim_neg_infty: shows "(\<lambda>i. cdf M (- real i)) \<longlonglongrightarrow> 0"
UP_cring: fixes R :: "('c, 'd) ring_scheme" assumes "UP_cring R" shows "cring (UP R)"
UNIF1_unifD: fixes \<sigma> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset"   and E' :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset" assumes "UNIF1 \<sigma> E E'" shows "(E, E') \<in> unif"
simps_2: fixes uu :: "'a"   and v :: "'b \<times> 'a"   and va :: "('b \<times> 'a) list" shows "Dioid_Models_New.last (uu, v # va) = snd (List.last (v # va))"
extend_constrains_project_set: fixes F :: "'a program"   and A :: "'c set"   and B :: "'c set" assumes "extend h F \<in> A co B" shows "F \<in> project_set h A co project_set h B"
minus_eq: fixes m :: "nat"   and k :: "nat" shows "(m - k = m) = (k = 0 \<or> m = 0)"
select_convs_1: fixes lens_get :: "'e \<Rightarrow> 'f"   and lens_put :: "'e \<Rightarrow> 'f \<Rightarrow> 'e"   and more :: "'g" shows "get\<^bsub>\<lparr>lens_get = lens_get, lens_put = lens_put, \<dots> = more\<rparr>\<^esub> = lens_get"
maxim_Int: fixes A :: "'b set"   and B :: "'b set" assumes "finite A"   and "A \<noteq> {}"   and "A \<subseteq> Field s"   and "s.maxim A \<in> B" shows "s.maxim (A \<inter> B) = s.maxim A"
StatImp: fixes \<Psi>' :: "'b" shows "\<Psi>' \<rhd> Q \<lessapprox><Rel> P"
normal_intersection_hom: shows "group_hom (G\<lparr>carrier := S\<rparr>) (G\<lparr>carrier := H <#> S\<rparr> Mod H) ((#>) H)"
NH\<^sub>N_EH\<^sub>N: fixes n :: "nat" assumes "p \<in> {0<..<1}" shows "NH\<^sub>N n = ennreal (EH\<^sub>N n)"
callee_invariant_on_plus_oracle: fixes I :: "'s \<Rightarrow> bool"   and \<I>l :: "('a, 'b) \<I>"   and \<I>r :: "('c, 'd) \<I>" shows "callee_invariant_on (left \<oplus>\<^sub>O right) I (\<I>l \<oplus>\<^sub>\<I> \<I>r) = (callee_invariant_on left I \<I>l \<and> callee_invariant_on right I \<I>r)"
eq_strict_2: fixes y :: "'a" shows "eq\<cdot>\<bottom>\<cdot>y = \<bottom>"
fg_start_round: fixes A :: "('f, 'g, 'h, 'i) CHOAlgorithm_scheme"   and rho :: "nat \<Rightarrow> ('g, 'f, 'h) config"   and HOs :: "nat \<Rightarrow> 'f \<Rightarrow> 'f set"   and SHOs :: "nat \<Rightarrow> 'f \<Rightarrow> 'f set"   and coords :: "nat \<Rightarrow> 'f \<Rightarrow> 'f"   and p :: "'f"   and r :: "nat" assumes "fg_run A rho HOs SHOs coords" shows "round (rho (fg_start_round rho p r)) p = r"
leftmost_cons_terminal: fixes x :: "symbol"   and i :: "nat"   and xs :: "symbol list" assumes "is_terminal x" shows "leftmost i (x # xs) = (0 < i \<and> leftmost (i - 1) xs)"
simps_2: fixes pending_write :: "bool"   and x :: "'b memref"   and xs :: "'b memref list"   and A :: "nat set" shows "acquired pending_write (x # xs) A = (case x of Write\<^sub>s\<^sub>b volatile x xa xb A' L R W \<Rightarrow> if volatile then acquired True xs (if pending_write then A \<union> A' - R else A' - R) else acquired pending_write xs A | Ghost\<^sub>s\<^sub>b A' L R W \<Rightarrow> acquired pending_write xs (if pending_write then A \<union> A' - R else A) | _ \<Rightarrow> acquired pending_write xs A)"
distanceI: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node"   and x :: "nat" assumes "n -as\<rightarrow>\<^sub>\<iota>* n'"   and "length as = x"   and "\<forall>as'. n -as'\<rightarrow>\<^sub>\<iota>* n' \<longrightarrow> x \<le> length as'" shows "distance n n' x"
rep_eq: fixes x :: "float"   and xa :: "float" shows "real_of_float (x - xa) = real_of_float x - real_of_float xa"
YBnd: shows "Y \<subseteq> Bounded_vec Bnd"
nth_default_def: fixes dflt :: "'f"   and xs :: "'f list"   and n :: "nat" shows "nth_default dflt xs n = (if n < length xs then xs ! n else dflt)"
filter_empty_ivls_def: fixes X :: "'c" shows "filter_empty_ivls X = X"
simps_1: fixes C :: "char list"   and Cs :: "char list list"   and b :: "bool"   and e :: "'a exp" shows "init_exp_of (INIT C (Cs,b) \<leftarrow> e) = \<lfloor>e\<rfloor>"
Int_space_eq2: fixes \<Omega> :: "'b set"   and M :: "'b set set"   and x :: "'b set" assumes "semiring_of_sets \<Omega> M"   and "x \<in> M" shows "x \<inter> \<Omega> = x"
card_insert_disjoint: fixes A :: "'e set"   and x :: "'e" assumes "finite A"   and "x \<notin> A" shows "card (insert x A) = Suc (card A)"
homeomorphic_path_connectedness: fixes S :: "'b set"   and T :: "'c set" assumes "S homeomorphic T" shows "path_connected S = path_connected T"
IH: assumes "distinct lC" shows "distinct (map ((\<cdot>\<^sub>v) c) lC)"
bal_i_insert_d: fixes x :: "'a"   and t :: "'a tree"   and dl :: "nat"   and t' :: "'a tree"   and dl' :: "nat" assumes "insert_d x (t, dl) = (t', dl')"   and "bal_i (size t + dl) (height t)" shows "bal_i (size t' + dl) (height t')"
distinct_2: fixes x2 :: "'a"   and x1 :: "'a" shows "Neg x2 \<noteq> Pos x1"
ord_eq_Eequiv_2: fixes v :: "i"   and x :: "\<kappa>"   and y :: "\<kappa>" shows "[x \<^bold>=\<^sub>E y \<^bold>\<rightarrow> y \<^bold>=\<^sub>E x in v]"
r_const_prim: fixes c :: "nat" shows "prim_recfn 1 (r_const c)"
wf_imp_irrefl: fixes r :: "('b \<times> 'b) set" assumes "wf r" shows "irrefl r"
octo_inverse_eq_cnj: fixes q :: "octo" assumes "norm q = 1" shows "inverse q = Octonions.cnj q"
n: shows "fg_start_round rho p (Suc r) = Suc n"
refl: fixes n :: "'node"   and S :: "'node set" assumes "valid_node n"   and "n \<in> S" shows "n \<in> wod_backward_slice S"
intros_41: fixes v :: "e"   and i :: "nat" assumes "is_const v" shows "\<lparr>[v, $Tee_local i]\<rparr> \<leadsto> \<lparr>[v, v, $Set_local i]\<rparr>"
mutilated_board_def: fixes m :: "nat"   and n :: "nat" shows "mutilated_board m n = below (2 * (m + 1)) \<times> below (2 * (n + 1)) - {(0, 0)} - {(2 * m + 1, 2 * n + 1)}"
le_inf_concat_start: fixes w :: "'item list"   and w\<^sub>1 :: "'item stream"   and w\<^sub>2 :: "'item stream" shows "(w @- w\<^sub>1 \<preceq>\<^sub>I w @- w\<^sub>2) = (w\<^sub>1 \<preceq>\<^sub>I w\<^sub>2)"
ss_422iii: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "is_p_fun y" shows "(x \<cdot> z ; y\<^sup>\<smile>) ; y = x ; y \<cdot> z"
foreach_body_def: fixes u :: "'b" shows "foreach_body u \<equiv> \<lambda>(v, d) (Q, \<pi>). if v = r then (Q, \<pi>) else case (Q v, \<pi> v) of (enat d', x) \<Rightarrow> if d < d' then (Q(v := enat d), \<pi>(v \<mapsto> u)) else (Q, \<pi>) | (\<infinity>, None) \<Rightarrow> (Q(v := enat d), \<pi>(v \<mapsto> u)) | (\<infinity>, Some xa) \<Rightarrow> (Q, \<pi>)"
ad: shows "adjoint (tensor_P (1\<^sub>m N) (mat_incr K)) = tensor_P (1\<^sub>m N) (adjoint (mat_incr K))"
word_rot_rl: fixes k :: "nat"   and v :: "'b word" shows "word_rotl k (word_rotr k v) = v"
ccSUP_empty: 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 f :: "'b \<Rightarrow> 'a" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (f ` {}) = bot"
scale_right_diff_distrib: fixes a :: "'a"   and x :: "'a"   and y :: "'a" shows "a * (x - y) = a * x - a * y"
ideD_1: fixes a :: "'a" assumes "hom.ide a" shows "hom.arr a"
aff_independent_finite: fixes B :: "'b set" assumes "\<not> affine_dependent B" shows "finite B"
disj_cancel_right: fixes x :: "'b" shows "x OR NOT x = - (1::'b)"
term_ok_subst_bv: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and T :: "typ"   and t :: "Core.term"   and x :: "variable" assumes "term_ok \<Theta> (Abs T t)" shows "term_ok \<Theta> (subst_bv (Fv x T) t)"
"1_2": shows "\<not> \<not> wordinterval_empty r"
MaxNonConflict_def: fixes ST :: "('d, 'e, 'f) status"   and T :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set" shows "MaxNonConflict ST T \<equiv> T \<subseteq> ET ST \<and> (\<forall>A\<in>SAs (HA ST). card (T \<inter> Delta A) \<le> 1) \<and> (\<forall>t\<in>ET ST. (t \<in> T) = (\<not> (\<exists>t'\<in>ET ST. HigherPriority (HA ST) (t', t))))"
dvd_def: fixes b :: "'c"   and a :: "'c" shows "(b dvd a) = (\<exists>k. a = b * k)"
prems: shows "\<nexists>w c. awalk w c w \<and> w \<in> set (awalk_verts u p) \<and> awalk_cost f c < 0"
eq: shows "\<forall>A\<in>sets M. \<integral>\<^sup>+ x. f x * indicator A x \<partial>M = \<integral>\<^sup>+ x. g x * indicator A x \<partial>M"
nprt_neg: 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 inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "lattice_ab_group_add.nprt zero inf (uminus x) = uminus (lattice_ab_group_add.pprt zero sup x)"
comp_of_ords_of_le_lt: fixes c :: "'c \<Rightarrow> 'c \<Rightarrow> order" shows "comp_of_ords (le_of_comp c) (lt_of_comp c) = c"
Chernoff_on_def: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" shows "Chernoff_on A f = (\<forall>B\<subseteq>A. \<forall>C\<subseteq>B. f B \<inter> C \<subseteq> f C)"
sys_xcpts_def: shows "sys_xcpts = set sys_xcpts_list"
divide: fixes e :: "'c \<times> 'c"   and H :: "'c pair_pre_digraph"   and w :: "'c"   and G :: "'c pair_pre_digraph" assumes "e \<in> parcs H"   and "w \<notin> pverts H"   and "??.Subdivision.pair_sd G H" shows "??.Subdivision.pair_sd G (subdivide H e w)"
prob_space_axioms: shows "prob_space (restrict_space lborel {0<..<1})"
elims: fixes x :: "int poly \<times> rat \<times> rat"   and y :: "rat" assumes "rai_ub x = y" obtains uu_ :: "int poly" and uv_ :: "rat" and r :: "rat" where "x = (uu_, uv_, r)" and "y = r"
less_sum_def: fixes x :: "'a + 'b"   and y :: "'a + 'b" shows "(x < y) = strict (\<le>) x y"
wf_below_def: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" shows "wf_below r A = wf (Restr r ((r\<^sup>*)\<inverse> `` A))"
dist_exp_i_1: fixes t :: "real" shows "cmod (exp (\<i> * complex_of_real t) - 1) = 2 * \<bar>sin (t / 2)\<bar>"
integrable_uniform_limit_real: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'c" assumes "\<And>e. 0 < e \<Longrightarrow> \<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}" shows "f integrable_on {a..b}"
univalent_union: fixes f_a :: "('d \<times> 'e) set"   and f_b :: "('d \<times> 'e) set" assumes "univalent f_a"   and "univalent f_b"   and "Domain f_a \<inter> Domain f_b = Domain (f_a \<inter> f_b)" shows "univalent (f_a \<union> f_b)"
store_barrier_pending_mod_privilege: fixes s' :: "'b sparc_state"   and b :: "bool"   and s :: "'b sparc_state" assumes "s' = store_barrier_pending_mod b s \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
addcol_mat: fixes A :: "'b mat"   and nr :: "nat"   and n :: "nat"   and k :: "nat"   and a :: "'b"   and l :: "nat" assumes "A \<in> carrier_mat nr n"   and "k < n" shows "addcol a l k A = A * addrow_mat n a k l"
invert_side_of_triangle_1: fixes h :: "'a \<times> 'a"   and f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "CC.arr h"   and "CC.comp f g = h"   and "CC.iso f" shows "CC.seq (CC.inv f) h \<and> g = CC.comp (CC.inv f) h"
submatrix_index: fixes i :: "nat"   and A :: "'b mat"   and I :: "nat set"   and j :: "nat"   and J :: "nat set" assumes "i < card {i. i < dim_row A \<and> i \<in> I}"   and "j < card {j. j < dim_col A \<and> j \<in> J}" shows "submatrix A I J $$ (i, j) = A $$ (pick I i, pick J j)"
assms_1: fixes x :: "'a" shows "PROTECT P x \<longrightarrow> P' x"
coprime_mult_cross_dvd_1: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and p :: "'a"   and q :: "'a"   and p' :: "'a"   and q' :: "'a" assumes "class.idom_gcd times one plus zero minus uminus gcd"   and "comm_monoid_mult.coprime times one p q"   and "times p' p = times q' q" shows "dvd.dvd times p q'"
nfv_And: fixes \<phi> :: "'b formula"   and \<psi> :: "'b formula" shows "nfv (And \<phi> \<psi>) = max (nfv \<phi>) (nfv \<psi>)"
IH: fixes f' :: "'a \<Rightarrow> 'a ttree" assumes "xs \<in> paths (substitute (f_nxt f T x) T (nxt t x \<otimes>\<otimes> f x))"   and "\<And>xa. xa \<in> A \<Longrightarrow> carrier (f_nxt f T x xa) \<subseteq> A"   and "carrier (nxt t x \<otimes>\<otimes> f x) \<subseteq> A"   and "\<And>xa. xa \<in> A \<Longrightarrow> f_nxt f T x xa = f' xa" shows "xs \<in> paths (substitute f' T (nxt t x \<otimes>\<otimes> f x))"
C: shows "n \<notin> bad_shakey \<inter> (bad_pwd \<union> bad_prikey) \<inter> (bad_id_pubkey \<union> bad_id_shak)"
distinct_1: fixes x11 :: "'a"   and x12 :: "'a process"   and x21 :: "'a process"   and x22 :: "'a process" shows "Action x11 x12 \<noteq> Choice x21 x22"
tendsto_min: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and X :: "'c \<Rightarrow> 'b"   and x :: "'b"   and net :: "'c filter"   and Y :: "'c \<Rightarrow> 'b"   and y :: "'b" assumes "class.linorder_topology less_eq less open"   and "filterlim X (topological_space.nhds open x) net"   and "filterlim Y (topological_space.nhds open y) net" shows "LIM x net. ord.min less_eq (X x) (Y x) :> topological_space.nhds open (ord.min less_eq x y)"
assms: shows "(cms, mem) \<leadsto>\<^bsub>k\<^esub> (cms', mem')"
homeomorphism_scaleR: fixes c :: "real" assumes "c \<noteq> 0" shows "homeomorphism UNIV UNIV ((*\<^sub>R) c) (\<lambda>x. x /\<^sub>R c)"
irreflp_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "irreflp R = (\<forall>a. \<not> R a a)"
neutral: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "\<forall>x\<in>A. g x = bot" shows "sup_monoid.sum g A = bot"
\<omega>_limit_set_in_compact_\<omega>_limit_set_contained: fixes K :: "'a set"   and x :: "'a" assumes "compact K"   and "K \<subseteq> X"   and "x \<in> X"   and "trapped_forward x K" shows "\<forall>y\<in>\<omega>_limit_set x. \<omega>_limit_set y \<subseteq> \<omega>_limit_set x"
gen_lossless_bind_gpvD2: fixes co :: "bool"   and \<I> :: "('d, 'e) \<I>"   and gpv :: "('g, 'd, 'e) gpv"   and f :: "'g \<Rightarrow> ('f, 'd, 'e) gpv"   and x :: "'g" assumes "gen_lossless_gpv co \<I> (gpv \<bind> f)"   and "x \<in> results_gpv \<I> gpv" shows "gen_lossless_gpv co \<I> (f x)"
assms_3: shows "x \<le> z \<squnion> y"
spectral_radius_max: fixes A :: "((complex, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and v :: "complex" assumes "eigen_value A v" shows "cmod v \<le> HMA_Connect.spectral_radius A"
finite_extension_incl: fixes K :: "'a set"   and xs :: "'a list" assumes "K \<subseteq> carrier R"   and "set xs \<subseteq> carrier R" shows "K \<subseteq> finite_extension K xs"
takeWhile_replicate: fixes f :: "'b \<Rightarrow> bool"   and len :: "nat"   and x :: "'b" shows "takeWhile f (replicate len x) = (if f x then replicate len x else [])"
lin_dep_submatrix_UNIV: fixes A :: "'a mat"   and nc :: "nat"   and I :: "nat set" assumes "A \<in> carrier_mat n nc"   and "lin_dep (set (cols A))"   and "distinct (cols (submatrix A I UNIV))" shows "module.lin_dep class_ring (module_vec TYPE('a) (card {i. i < n \<and> i \<in> I})) (set (cols (submatrix A I UNIV)))"
strict_order_lfp: 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 f :: "('c \<times> 'c) set \<Rightarrow> ('c \<times> 'c) set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "mono f"   and "\<And>R. strict_order R \<Longrightarrow> strict_order (f R)" shows "strict_order (lfp f)"
is_sub_seq_def: fixes a :: "'a"   and b :: "'a"   and x :: "'a list" shows "is_sub_seq a b x \<equiv> \<exists>n. Suc n < length x \<and> x ! n = a \<and> x ! Suc n = b"
remap_lang: fixes f :: "'Q1 \<Rightarrow> 'a" assumes "inj_on f a1.Q" shows "ta_lang (ta_remap f TA) = ta_lang TA"
t_de_morgan_var1: fixes x :: "'a"   and y :: "'a" shows "!(!x \<cdot> !y) = t x + t y"
map_iterator_linord_rule_P: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and it :: "('d \<Rightarrow> bool) \<Rightarrow> ('c \<times> 'e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and m :: "'c \<Rightarrow> 'e option"   and I :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and \<sigma>0 :: "'d"   and c :: "'d \<Rightarrow> bool"   and f :: "'c \<times> 'e \<Rightarrow> 'd \<Rightarrow> 'd"   and P :: "'d \<Rightarrow> bool" assumes "class.linorder less_eq less"   and "linorder.set_iterator_map_linord less_eq it (map_to_set m)"   and "I (dom m) \<sigma>0"   and "\<And>k v it \<sigma>. \<lbrakk>c \<sigma>; k \<in> it; m k = Some v; it \<subseteq> dom m; I it \<sigma>; \<And>k'. k' \<in> it \<Longrightarrow> less_eq k k'; \<And>k'. k' \<in> dom m - it \<Longrightarrow> less_eq k' k\<rbrakk> \<Longrightarrow> I (it - {k}) (f (k, v) \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> it. \<lbrakk>it \<subseteq> dom m; it \<noteq> {}; \<not> c \<sigma>; I it \<sigma>; \<And>k k'. \<lbrakk>k \<in> dom m - it; k' \<in> it\<rbrakk> \<Longrightarrow> less_eq k k'\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it c f \<sigma>0)"
map_bset_def: shows "map_bset \<equiv> (id ---> set_bset ---> Abs_bset) (`)"
set_2: fixes x21 :: "('a, 'b, 'c) expr"   and x22 :: "('a, 'b, 'c) expr" shows "RID\<^sub>E (Apply x21 x22) = RID\<^sub>E x21 \<union> RID\<^sub>E x22"
bounded_iff: fixes b :: "'b multiset"   and c :: "'b multiset"   and a :: "'b multiset" shows "(b \<union># c \<subseteq># a) = (b \<subseteq># a \<and> c \<subseteq># a)"
swap_def: fixes n :: "nat"   and xs :: "'c list" shows "swap n xs = (if Suc n < length xs then xs[n := xs ! Suc n, Suc n := xs ! n] else xs)"
intro: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and bisim :: "'b \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool" assumes "multithreaded final1 r1"   and "multithreaded final2 r2"   and "FWbisimulation_axioms final1 r1 final2 r2 bisim" shows "FWbisimulation final1 r1 final2 r2 bisim"
grid_base_dim_add: fixes ds' :: "nat set"   and ds :: "nat set"   and b :: "(nat \<times> int) list"   and dm :: "nat"   and p :: "(nat \<times> int) list" assumes "ds' \<subseteq> ds"   and "b \<in> sparsegrid' dm"   and "p \<in> grid (base ds' b) ds'" shows "p \<in> grid (base ds b) ds"
singleton_matrix_def: fixes j :: "nat"   and i :: "nat"   and a :: "'b" shows "singleton_matrix j i a \<equiv> Abs_matrix (\<lambda>m n. if j = m \<and> i = n then a else (0::'b))"
OclIf_true': fixes \<tau> :: "'a state \<times> 'a state"   and P :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and B\<^sub>1 :: "'a state \<times> 'a state \<Rightarrow> 'b"   and B\<^sub>2 :: "'a state \<times> 'a state \<Rightarrow> 'b" assumes "\<tau> \<Turnstile> P" shows "(if P then B\<^sub>1 else B\<^sub>2 endif) \<tau> = B\<^sub>1 \<tau>"
A: fixes x :: "'a" shows "case lookup m x of None \<Rightarrow> True | Some v' \<Rightarrow> k = x \<or> P x v'"
merge_Err: fixes ss :: "(nat \<times> 'a) list"   and c :: "'a list"   and pc :: "nat" assumes "snd ` set ss \<subseteq> A" shows "merge c pc ss \<top> = \<top>"
ideD_1: fixes a :: "'j" assumes "J.ide a" shows "J.arr a"
eventually_def: fixes F :: "(nat \<Rightarrow> 'c) \<Rightarrow> bool" shows "\<diamond>F \<equiv> \<not> \<box>\<not> F"
flowD_finite: fixes \<Delta> :: "('a, 'b) network_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and e :: "'a \<times> 'a" assumes "countable_network \<Delta>"   and "flow \<Delta> g" shows "g e \<noteq> \<top>"
someI: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "P x" shows "P (Eps P)"
4: shows "Ind (set [a]) (set u\<^sub>1)"
coeff_map_poly: fixes f :: "'e \<Rightarrow> 'd"   and p :: "'e poly"   and n :: "nat" assumes "f (0::'e) = (0::'d)" shows "coeff (map_poly f p) n = f (coeff p n)"
conc_trans_additional_5: fixes A :: "'e nres"   and B :: "'e nres"   and C :: "'e nres" assumes "A \<le> \<Down> Id B"   and "B \<le> C" shows "A \<le> C"
is_extensional: fixes f :: "'a \<times> 'a" assumes "\<not> CC.arr f" shows "\<Pi>o\<pi>_\<delta>o\<Pi>.map f = null"
eval_prefs_from_table_3: fixes i :: "'c"   and j :: "'c"   and y :: "'d set list"   and xs :: "('c \<times> 'd set list) list" assumes "i \<noteq> j" shows "prefs_from_table ((j, y) # xs) i = prefs_from_table xs i"
sum_mono: fixes K :: "'c set"   and f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a" assumes "\<And>i. i \<in> K \<Longrightarrow> f i \<le> g i" shows "comm_monoid_add.sum (\<squnion>) bot f K \<le> comm_monoid_add.sum (\<squnion>) bot g K"
Zp_residue_a_inv_2: fixes a :: "nat \<Rightarrow> int"   and k :: "nat" assumes "a \<in> carrier Zp" shows "(\<ominus> a) k = - a k mod p ^ k"