hcju/isabelleps huggingface dataset:
simps_1: fixes uu :: "'a \<Rightarrow> 'b fundef option"   and \<Sigma> :: "Unboxed.type option list" shows "Subx.sp uu [] \<Sigma> = Ok \<Sigma>"
trcl_rev_cons: fixes c :: "'f"   and w :: "'g list"   and ch :: "'f"   and T :: "('f \<times> 'g \<times> 'f) set"   and e :: "'g"   and c' :: "'f" assumes "(c, w, ch) \<in> trcl T"   and "(ch, e, c') \<in> T" shows "(c, w @ [e], c') \<in> trcl T"
assms_1: shows "eoutKnowCorrect C (sKS m)"
ccpo_Sup_singleton: fixes Sup :: "'c set \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.ccpo Sup less_eq less" shows "Sup {x} = x"
Max_ge_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and x :: "'a" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "less_eq x (linorder.Max less_eq A) = (\<exists>a\<in>A. less_eq x a)"
mat_to_cols_list_times_mat: fixes A :: "complex Matrix.mat"   and B :: "complex Matrix.mat" assumes "dim_col A = dim_row B"   and "0 < dim_col A" shows "mat_to_cols_list (A * B) = mat_mult (mult.row_length (mat_to_cols_list A)) (mat_to_cols_list A) (mat_to_cols_list B)"
Conjuctive_Idempotent: fixes H :: "'d upred \<Rightarrow> 'd upred" assumes "Conjunctive H" shows "Idempotent H"
sup_absorb2: fixes sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.semilattice_sup sup less_eq less"   and "less_eq x y" shows "sup x y = y"
simps_1: fixes uu :: "'b \<Rightarrow> bool" shows "has_disc uu MatchAny = False"
simps: fixes \<delta> :: "'d \<Rightarrow> 'e \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'e"   and i :: "nat" shows "run\<^sub>t \<delta> q\<^sub>0 w i = (run \<delta> q\<^sub>0 w i, w i, run \<delta> q\<^sub>0 w (Suc i))"
is_rbt_def: fixes t :: "('a, 'b) rbt" shows "is_rbt t = (cl_inv t \<and> bd_inv t)"
r_unbdd: fixes x :: "real" shows "\<exists>n. x < r n"
scale: fixes r :: "'a"   and b :: "'b" shows "f (s1 r b) = r * f b"
order_trans: fixes y :: "('a, 'b) Term.term"   and x :: "('a, 'b) Term.term"   and z :: "('a, 'b) Term.term" assumes "y \<unrhd> x"   and "z \<unrhd> y" shows "z \<unrhd> x"
distinct_zipI1: fixes xs :: "'g list"   and ys :: "'h list" assumes "distinct xs" shows "distinct (zip xs ys)"
If': fixes b :: "bexp"   and l :: "nat"   and c\<^sub>1 :: "com"   and c\<^sub>2 :: "com" assumes "sec b \<le> l"   and "l \<turnstile>' c\<^sub>1"   and "l \<turnstile>' c\<^sub>2" shows "l \<turnstile>' IF b THEN c\<^sub>1 ELSE c\<^sub>2"
IO: fixes out :: "'h"   and \<I> :: "('h, 'i) \<I>"   and D :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and c :: "'i \<Rightarrow> 'j"   and c' :: "'i \<Rightarrow> 'k"   and A :: "'l \<Rightarrow> 'm \<Rightarrow> bool" assumes "out \<in> outs_\<I> \<I>"   and "\<And>input. input \<in> responses_\<I> \<I> out \<Longrightarrow> D (c input) (c' input)" shows "eq_\<I>_generat A \<I> D (IO out c) (IO out c')"
\<tau>move2_FAccD: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and e :: "'addr expr1"   and F :: "String.literal"   and D :: "String.literal"   and pc :: "nat"   and xcp :: "'addr option" assumes "\<tau>move2 P h stk (e\<bullet>F{D}) pc xcp"   and "pc < length (compE2 e)" shows "\<tau>move2 P h stk e pc xcp"
tLength_tMap: fixes f :: "'c \<Rightarrow> 'b"   and t :: "'c Trace" shows "tLength (tMap f t) = tLength t"
no_better_thansubset_rel: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "rational_preference carrier relation"   and "x \<in> carrier"   and "y \<in> carrier"   and "no_better_than y carrier relation \<subseteq> no_better_than x carrier relation" shows "x \<succeq>[relation] y"
primitive_prod_list: fixes fs :: "'a poly list"   and f :: "'a poly" assumes "primitive (prod_list fs)"   and "f \<in> set fs" shows "primitive f"
same_values_def: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and f :: "'b trm \<Rightarrow> 'b trm" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.same_values f = (\<lambda>x y. f x = f y)"
less_oLimitD: fixes x :: "ordinal"   and f :: "nat \<Rightarrow> ordinal" assumes "x < oLimit f" shows "\<exists>n. x < f n"
list_of_vec_nth: fixes i :: "nat"   and x :: "'c vec" assumes "i < dim_vec x" shows "list_of_vec x ! i = x $ i"
PTrans: shows "P \<Longrightarrow>a<\<nu>x> \<prec> P'"
col_per_perp: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> B"   and "B \<noteq> C"   and "D \<noteq> C"   and "Tarski_neutral_dimensionless.Col Bet B C D"   and "Tarski_neutral_dimensionless.Per Bet Cong A B C" shows "Tarski_neutral_dimensionless.Perp Bet Cong C D A B"
mapping_empty_choose_def: shows "mapping_empty_choose = Mapping.empty"
elims_2: fixes \<alpha> :: "real"   and x :: "'b tree" assumes "WBT0 \<alpha>"   and "WBT0.wbt \<alpha> x" obtains "x = \<langle>\<rangle>"   | l :: "'b tree" and uu_ :: "'b" and r :: "'b tree" where "x = \<langle>l, uu_, r\<rangle>" and "WBT0.wbt \<alpha> l \<and> WBT0.wbt \<alpha> r \<and> (let ratio = real (size1 l) / real (size1 l + size1 r) in \<alpha> \<le> ratio \<and> ratio \<le> 1 - \<alpha>)"
nth_drop: fixes n :: "nat"   and xs :: "'b list"   and i :: "nat" assumes "n \<le> \<^bold>|xs\<^bold>|" shows "drop n xs ! i = xs ! (n + i)"
nextnat_def: fixes s :: "nat \<Rightarrow> 'b" shows "nextnat s \<equiv> if emptyseq s then 0 else LEAST i. s i \<noteq> s 0"
simps_3: fixes vT :: "variable \<times> typ"   and n :: "nat"   and f :: "Core.term"   and u :: "Core.term" shows "bind_fv2 vT n (f $ u) = bind_fv2 vT n f $ bind_fv2 vT n u"
unfold_congs_3: fixes r :: "('c, 'd, 'e) ipurge_rec_scheme"   and r' :: "('c, 'd, 'e) ipurge_rec_scheme"   and v' :: "'d"   and f :: "'d \<Rightarrow> 'd"   and f' :: "'d \<Rightarrow> 'd" assumes "r = r'"   and "Dom r' = v'"   and "\<And>v. v = v' \<Longrightarrow> f v = f' v" shows "Dom_update f r = Dom_update f' r'"
sconst_alt: fixes s :: "'c stream"   and x :: "'c" shows "(s = sconst x) = (sset s = {x})"
times_in_intervalE: fixes xy :: "'b"   and X :: "'b interval"   and Y :: "'b interval" assumes "xy \<in>\<^sub>i X * Y" obtains x :: "'b" and y :: "'b" where "xy = x * y" and "x \<in>\<^sub>i X" and "y \<in>\<^sub>i Y"
rep_eq: fixes x :: "'g \<Rightarrow> ('f, 'e) uexpr"   and xa :: "('g, 'e) uexpr" shows "\<lbrakk>msubst x xa\<rbrakk>\<^sub>e = (\<lambda>b. \<lbrakk>x (\<lbrakk>xa\<rbrakk>\<^sub>e b)\<rbrakk>\<^sub>e b)"
assms_2: shows "E \<subseteq> Pow \<Omega>"
wt_args: shows "\<lparr>prg = G, cls = accC, lcl = L\<rparr>,empty_dt\<Turnstile>ps\<Colon>\<doteq>pTs"
simps_2: fixes i :: "'a"   and l :: "'a factor_tree"   and r :: "'a factor_tree" shows "sub_trees (Factor_Node i l r) = insert (Factor_Node i l r) (sub_trees l \<union> sub_trees r)"
last_message_conv: fixes m :: "'a message_af"   and xs :: "'a message_af list" assumes "m \<noteq> NoMsg" shows "(last_message xs = m) = (\<exists>i<length xs. xs ! i = m \<and> (\<forall>j<length xs. i < j \<longrightarrow> xs ! j = NoMsg))"
k_def: shows "k = Cn 1 h [m, recf.Id 1 0]"
gSubstInpPresGWlsInp_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'e, 'f) Recursion.model_scheme" shows "gSubstInpPresGWlsInp MOD \<equiv> \<forall>ys y Y' Y delta inp' inp. wls (asSort ys) Y' \<and> gWls MOD (asSort ys) Y \<and> wlsInp delta inp' \<and> gWlsInp MOD delta inp \<longrightarrow> gWlsInp MOD delta (gSubstInp MOD ys Y' Y y inp' inp)"
sorted_list_of_set_3: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "class.linorder less_eq less" shows "distinct (linorder.sorted_list_of_set less_eq A)"
sublist_refl: fixes l :: "'a list" shows "subseq l l"
module_hom_def: fixes s1 :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and s2 :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> 'e" shows "module_hom s1 s2 f \<equiv> module s1 \<and> module s2 \<and> module_hom_axioms s1 s2 f"
Ord_mem_iff_less_TC: fixes l :: "V"   and k :: "V" assumes "Ord l" shows "(k \<in> elts l) = (k \<sqsubset> l)"
inducts: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x1 :: "'a"   and x2 :: "'a"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "r\<^sup>+\<^sup>+ x1 x2"   and "\<And>a b. r a b \<Longrightarrow> P a b"   and "\<And>a b c. \<lbrakk>r\<^sup>+\<^sup>+ a b; P a b; r b c\<rbrakk> \<Longrightarrow> P a c" shows "P x1 x2"
incl_in_img_cod: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and f :: "'b" assumes "set_category S \<AA>"   and "partial_magma.arr S f" shows "set_category.incl_in S \<AA> (set_category.img S \<AA> f) (partial_magma.cod S f)"
exhaust: fixes y :: "'b array" obtains x :: "'b list" where "y = Array x"
join_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> y \<squnion> z = x \<squnion> (y \<squnion> z)"
1: shows "lselect {m. k + m \<in> s} (ldropn k w) = <>"
pre_Bisim_def: fixes as :: "'b list"   and R :: "'b rexp set"   and S :: "'b rexp set" shows "pre_Bisim as R S = (\<lambda>(ws, ps). (R, S) \<in> set ws \<union> set ps \<and> (\<forall>(R, S)\<in>set ws \<union> set ps. Atoms R \<union> Atoms S \<subseteq> set as) \<and> (\<forall>(R, S)\<in>set ps. Nullable R = Nullable S \<and> (\<forall>a\<in>set as. (Pderiv a R, Pderiv a S) \<in> set ps \<union> set ws)))"
T_quasi_preserves_null_2: fixes A :: "'a set"   and n :: "nat" assumes "A \<in> sets M" shows "(A \<in> null_sets M) = ((T ^^ n) --` A \<in> null_sets M)"
ex_se_star_succ: fixes rv :: "'a \<times> nat"   and ls :: "('b, 'c) label list" assumes "RedBlack prb"   and "rv \<in> red_vertices prb"   and "finite_labels ls" shows "\<exists>c'. se_star (confs prb rv) ls c'"
le1: shows "adjoint M1 * Q * M1 \<le>\<^sub>L adjoint M1 * M1"
fmap_fmap: fixes f :: "'f \<rightarrow> 'd"   and g :: "'g \<rightarrow> 'f"   and xs :: "'g\<cdot>'e" shows "fmap\<cdot>f\<cdot>(fmap\<cdot>g\<cdot>xs) = fmap\<cdot>(\<Lambda> x. f\<cdot>(g\<cdot>x))\<cdot>xs"
lunit_eqI: fixes f :: "'c"   and a :: "'c" assumes "\<guillemotleft>f : \<I>\<^sub>C \<otimes>\<^sub>C a \<rightarrow>\<^sub>C a\<guillemotright>"   and "\<I>\<^sub>C \<otimes>\<^sub>C f = (\<iota>\<^sub>C \<otimes>\<^sub>C a) \<cdot>\<^sub>C C.assoc' \<I>\<^sub>C \<I>\<^sub>C a" shows "f = \<l>\<^sub>C[a]"
TyAll_canonical: fixes v :: "trm"   and T\<^sub>1 :: "POPLmarkRecord.type"   and T\<^sub>2 :: "POPLmarkRecord.type" assumes "[] \<turnstile> v : (\<forall><:T\<^sub>1. T\<^sub>2)"   and "v \<in> value" shows "\<exists>t S. v = (\<lambda><:S. t)"
new_tv_list: fixes n :: "nat"   and x :: "'b list" shows "new_tv n x = (\<forall>y\<in>set x. new_tv n y)"
pp_eq_iff: fixes s :: "('a, 'b) pp"   and t :: "('a, 'b) pp" shows "(s = t) = (lookup_pp s = lookup_pp t)"
induct: fixes P :: "'a pfm \<Rightarrow> bool"   and pfm :: "'a pfm" assumes "P \<^bold>\<bottom>\<^sub>!"   and "\<And>x. P (Pro\<^sub>! x)"   and "\<And>x1 x2a. \<lbrakk>P x1; P x2a\<rbrakk> \<Longrightarrow> P (x1 \<^bold>\<or>\<^sub>! x2a)"   and "\<And>x1 x2a. \<lbrakk>P x1; P x2a\<rbrakk> \<Longrightarrow> P (x1 \<^bold>\<and>\<^sub>! x2a)"   and "\<And>x1 x2a. \<lbrakk>P x1; P x2a\<rbrakk> \<Longrightarrow> P (x1 \<^bold>\<longrightarrow>\<^sub>! x2a)"   and "\<And>x1 x2a. P x2a \<Longrightarrow> P (K\<^sub>! x1 x2a)"   and "\<And>x1 x2a. \<lbrakk>P x1; P x2a\<rbrakk> \<Longrightarrow> P ([x1]\<^sub>! x2a)" shows "P pfm"
X_deX: fixes a :: "vbl" shows "X (deX a) = a"
dvd_mult: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a dvd c" shows "a dvd b * c"
enat_equalityI: fixes m :: "enat"   and n :: "enat"   and X :: "(enat \<times> enat) set" assumes "(m, n) \<in> X"   and "\<And>m n. (m, n) \<in> X \<Longrightarrow> m = 0 \<and> n = 0 \<or> (\<exists>m' n'. m = eSuc m' \<and> n = eSuc n' \<and> ((m', n') \<in> X \<or> m' = n'))" shows "m = n"
memoization_correct_1: fixes dp\<^sub>T :: "'a \<Rightarrow> ('b, 'c) state"   and x :: "'a"   and v :: "'c"   and m :: "'b" assumes "consistentDP dp\<^sub>T"   and "run_state (dp\<^sub>T x) empty = (v, m)" shows "dp x = v"
B'_2: shows "B' \<in> set (tr\<^sub>p\<^sub>c S D)"
R0_def: shows "R0 = {(sp1, sp2). \<exists>\<iota>' \<iota>'' V' V''. sp1 = [spawn\<^bsub>\<iota>'\<^esub> V'] \<and> sp2 = [spawn\<^bsub>\<iota>''\<^esub> V''] \<and> \<iota>' \<notin> set (PPV V) \<and> \<iota>'' \<notin> set (PPV V) \<and> (V', V'') \<in> R}"
length_1_calculate_data_satisfies_properties: fixes p :: "real poly"   and qs :: "real poly list" assumes "p \<noteq> 0"   and "length qs = 1"   and "\<forall>q. List.member qs q \<longrightarrow> coprime p q" shows "satisfies_properties p qs (get_subsets (calculate_data p qs)) (get_signs (calculate_data p qs)) (get_matrix (calculate_data p qs))"
Enc_keys_clean_analz: fixes X :: "msg"   and K :: "msg"   and S :: "msg set" assumes "Enc X K \<in> analz S"   and "Enc_keys_clean S" shows "K \<in> range LtK \<union> payload"
wfP_wf_eq: fixes r :: "('b \<times> 'b) set" shows "wfP (\<lambda>x y. (x, y) \<in> r) = wf r"
K_3: fixes k :: "('a \<Rightarrow> 'b) set" assumes "k \<in> K" shows "\<exists>X. k = Pi\<^sub>E UNIV X \<and> (\<forall>i. open (X i)) \<and> finite {i. X i \<noteq> UNIV}"
lscan_lfinite: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and w :: "'c llist"   and a :: "'b" shows "lfinite (lscan f w a) = lfinite w"
Weierstrass_l2_trig_polynomial: fixes f :: "real \<Rightarrow> real"   and e :: "real" assumes "f square_integrable {- pi..pi}"   and "0 < e" obtains n :: "nat" and a :: "nat \<Rightarrow> real" and b :: "nat \<Rightarrow> real" where "l2norm {- pi..pi} (\<lambda>x. f x - (\<Sum>k\<le>n. a k * sin (real k * x) + b k * cos (real k * x))) < e"
min_height_size1_log_if_incomplete: fixes t :: "'b tree" assumes "\<not> Tree.complete t" shows "real (min_height t) < log 2 (real (size1 t))"
sngs'_def: shows "sngs' = (\<lambda>n. complex_of_real (2 * pi * real n) * \<i>) ` {0<..N}"
mcont_contD: fixes luba :: "'f set \<Rightarrow> 'f"   and orda :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and lubb :: "'g set \<Rightarrow> 'g"   and ordb :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'g"   and Y :: "'f set" assumes "mcont luba orda lubb ordb f"   and "Complete_Partial_Order.chain orda Y"   and "Y \<noteq> {}" shows "f (luba Y) = lubb (f ` Y)"
arr_cod_iff_arr: fixes f :: "'a1" shows "C1.arr (C1.cod f) = C1.arr f"
vars_for_consts_sub_consts: fixes cs :: "'c list"   and p :: "('c, 'd) form" assumes "closed (length cs) p"   and "list_all (\<lambda>c. new c p) cs"   and "distinct cs" shows "vars_for_consts (sub_consts cs p) cs = p"
isSucc_def: fixes i :: "'a" shows "isSucc i \<equiv> \<exists>j. local.aboveS j \<noteq> {} \<and> i = succ j"
pigeonhole_infinite_rel: fixes A :: "'b set"   and B :: "'c set"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "infinite A"   and "finite B"   and "\<forall>a\<in>A. \<exists>b\<in>B. R a b" shows "\<exists>b\<in>B. infinite {a \<in> A. R a b}"
filter_M_not_more_elements: fixes h :: "'x"   and P :: "'z \<Rightarrow> ('x, 'y, bool) prog"   and xs :: "'z list"   and ys :: "'z list"   and x :: "'z" assumes "h \<turnstile> filter_M P xs \<rightarrow>\<^sub>r ys"   and "\<And>x. x \<in> set xs \<Longrightarrow> pure (P x) h"   and "x \<in> set ys" shows "x \<in> set xs"
intro: fixes x :: "'d"   and S :: "'d set" assumes "x \<notin> S" shows "iterative S x"
sup_bot_left: fixes sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bot :: "'c"   and x :: "'c" assumes "class.bounded_semilattice_sup_bot sup less_eq less bot" shows "sup bot x = x"
whisker_left: fixes f :: "('a S.E.term, 'a) S.arr"   and \<nu> :: "('a S.E.term, 'a) S.arr"   and \<mu> :: "('a S.E.term, 'a) S.arr" assumes "S.ide f"   and "S.seq \<nu> \<mu>" shows "f \<star>\<^sub>S \<nu> \<cdot>\<^sub>S \<mu> = (f \<star>\<^sub>S \<nu>) \<cdot>\<^sub>S (f \<star>\<^sub>S \<mu>)"
min_dist_z_iff: fixes v :: "nat"   and v' :: "nat" assumes "connected v v'" shows "(min_dist v v' = 0) = (v' = v)"
preserves_hom: fixes f :: "'j"   and a :: "'j"   and b :: "'j" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>J b\<guillemotright>" shows "\<guillemotleft>\<chi> f : \<chi>.A.map a \<rightarrow>\<^sub>C E b\<guillemotright>"
exp_strong_exhaust: fixes y :: "exp"   and c :: "'a" obtains var :: "Vars.var" where "y = Var var__"   | exp :: "exp" and var :: "Vars.var" where "y = App exp__ vara__"   | \<Gamma> :: "(Vars.var \<times> exp) list" and exp :: "exp" where "atom ` domA \<Gamma> \<sharp>* c" and "y = Let \<Gamma> expa__"   | var :: "Vars.var" and exp :: "exp" where "{atom var} \<sharp>* c" and "y = Lam [var]. exp"   | b :: "bool" where "y = Bool b"   | scrut :: "exp" and e1 :: "exp" and e2 :: "exp" where "y = (scrut ? e1 : e2)"
basis_in_card: fixes \<E> :: "'a set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set" assumes "\<E> \<subseteq> carrier"   and "basis_in \<E> B\<^sub>1"   and "basis_in \<E> B\<^sub>2" shows "card B\<^sub>1 = card B\<^sub>2"
Fix_lemma: fixes \<phi> :: "((Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool) \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes "Monotone \<phi>" shows "\<phi> (FIX \<phi>) = FIX \<phi>"
append_Nil: fixes ys :: "'e list" shows "append [] ys = ys"
return_def: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem" shows "IFC_def.return prob = ifc_problem.return prob"
w_values_new_actionD: fixes ad :: "'a"   and hT :: "htype"   and obs :: "('a, 'b) obs_event action list"   and al :: "addr_loc"   and P :: "'c prog"   and vs0 :: "'a \<times> addr_loc \<Rightarrow> 'a val set" assumes "NormalAction (NewHeapElem ad hT) \<in> set obs"   and "(ad, al) \<in> action_loc_aux P (NormalAction (NewHeapElem ad hT))" shows "addr_loc_default P hT al \<in> w_values P vs0 obs (ad, al)"
subcls1_class: fixes C :: "char list"   and D :: "char list"   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" assumes "P \<turnstile> C \<prec>\<^sup>1 D" shows "is_class P C"
sig_red_single_top_tail_cases: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and f :: "'t \<Rightarrow>\<^sub>0 'b"   and t :: "'a" shows "sig_red_single sing_reg (\<preceq>) p q f t = (sig_red_single sing_reg (=) p q f t \<or> sig_red_single sing_reg (\<prec>) p q f t)"
simps: fixes p1 :: "int poly"   and l1 :: "rat"   and r1 :: "rat"   and p2 :: "int poly"   and l2 :: "rat"   and r2 :: "rat" shows "add_1 (p1, l1, r1) (p2, l2, r2) = select_correct_factor_int_poly (tighten_poly_bounds_binary p1 p2) (\<lambda>((l1, r1, sr1), l2, r2, sr2). (l1 + l2, r1 + r2)) ((l1, r1, sgn (ipoly p1 r1)), l2, r2, sgn (ipoly p2 r2)) (poly_add p1 p2)"
stack_reduction_fresh: fixes k :: "stack"   and k' :: "stack"   and x :: "name" assumes " k \<mapsto> k' "   and "x \<sharp> k" shows "x \<sharp> k'"
m_2: shows "m \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t \<A>)"
minus_le_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq (uminus a) b = less_eq (uminus b) a"
pth_c_1: fixes c :: "real"   and x :: "'a"   and d :: "real"   and y :: "'a" shows "c *\<^sub>R x + d *\<^sub>R y \<equiv> d *\<^sub>R y + c *\<^sub>R x"
berlekamp_basis_irreducible: fixes f :: "'b mod_ring poly"   and n :: "nat"   and us :: "'b mod_ring poly list"   and u :: "'b mod_ring poly" assumes "square_free f"   and "n = length (berlekamp_basis f)"   and "monic f"   and "f = prod_list us"   and "length us = n"   and "\<And>u. u \<in> set us \<Longrightarrow> 0 < degree u"   and "u \<in> set us" shows "irreducible u"
separated_by_this_wall_gf: fixes C :: "'a set"   and D :: "'a set" assumes "separated_by {g \<turnstile> folding_g.\<C>, f \<turnstile> folding_g.\<C>} C D"   and "C \<in> g \<turnstile> folding_g.\<C>" shows "D \<in> f \<turnstile> folding_g.\<C>"
assms: fixes a :: "'a" shows "(t a e)[x::=y] = t a e[x::=y]"
Sup_option_def: fixes A :: "'b option set" shows "\<Squnion> A = (if A = {} \<or> A = {None} then None else Some (\<Squnion> Option.these A))"
order_iff: fixes a :: "'c"   and b :: "'c" shows "(a \<le> b) = (a = inf a b)"
dropWhile_eq_Cons_conv: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "(dropWhile P xs = y # ys) = (xs = takeWhile P xs @ y # ys \<and> \<not> P y)"
\<Gamma>_in_hom: fixes a :: "'c" assumes "C.obj a" shows "\<guillemotleft>\<Delta> a : \<sigma>\<^sub>0 a \<Rightarrow>\<^sub>D \<tau>\<^sub>0 a\<guillemotright>"
is_ipd_def: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and y :: "'a"   and x :: "'a" shows "IFC_def.is_ipd prob y x = (x \<noteq> y \<and> IFC_def.is_pd prob y x \<and> (\<forall>z. z \<noteq> x \<and> IFC_def.is_pd prob z x \<longrightarrow> IFC_def.is_pd prob z y))"
no_subcls_Object: fixes D :: "qtname"   and G :: "prog"   and P :: "bool" assumes "G\<turnstile>Object\<prec>\<^sub>C D" shows "P"
eq: shows "smult c p = 0 \<longrightarrow> c * d = (0::'a) \<and> bs = []"
map_1: fixes f :: "'e \<Rightarrow> 'd" shows "map f [] = []"
assms: shows "length (bc n) < length (bc (Suc n))"
heads_Cons: fixes x :: "bool"   and xs :: "bool list" shows "heads (x # xs) = (if x then heads xs + 1 else heads xs)"
supp_finsert: fixes x :: "'a"   and S :: "'a fset" shows "supp (finsert x S) = supp x \<union> supp S"
new_element_get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and ptr :: "(_) document_ptr"   and getter :: "('n, 'l, 'm, 'u option) RDocument_scheme \<Rightarrow> 'w" assumes "h \<turnstile> new_element \<rightarrow>\<^sub>h h'" shows "preserved (get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h h'"
poslsts_iff: fixes s :: "'b llist"   and A :: "'b set" shows "(s \<in> A\<^sup>\<spadesuit>) = (s \<in> A\<^sup>\<infinity> \<and> s \<noteq> LNil)"
iT_Mult_not_empty: fixes I :: "nat set"   and k :: "nat" assumes "I \<noteq> {}" shows "I \<otimes> k \<noteq> {}"
assms_1: shows "\<forall>x. g (f x) = x"
card_of_Fpow_cong: fixes A :: "'c set"   and B :: "'d set" assumes "|A| =o |B|" shows "|Fpow A| =o |Fpow B|"
monomial_prod_sum: fixes a :: "'e \<Rightarrow> 'c"   and I :: "'e set"   and c :: "'e \<Rightarrow> 'd" shows "monomial (prod c I) (sum a I) = (\<Prod>i\<in>I. monomial (c i) (a i))"
assms_5: shows "s\<^sub>1 \<noteq> []"
class_of_def: fixes U :: "'a set"   and s :: "'b" shows "\<lfloor>U, s\<rfloor> \<equiv> equivalence.Class (Sigma I \<FF>) {(x, y). x \<sim> y} (U, s)"
NO_MATCH_def: fixes pat :: "'b"   and val :: "'c" shows "NO_MATCH pat val \<equiv> True"
in_homE: fixes f :: "'a \<times> 'a \<times> 'a"   and a :: "'a \<times> 'a \<times> 'a"   and b :: "'a \<times> 'a \<times> 'a" assumes "CCC.in_hom f a b" obtains "\<guillemotleft>fst f : fst a \<rightarrow> fst b\<guillemotright>" and "CC.in_hom (snd f) (snd a) (snd b)"
indets_uminus: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "indets (- p) = indets p"
fps_unit_factor_mult_unit_factor_1: fixes f :: "'a fps"   and g :: "'a fps" shows "unit_factor (f * unit_factor g) = unit_factor (f * g)"