hcju/isabelleps huggingface dataset:
det_echelon_form_of_det': fixes bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and A :: "(('b, 'c) vec, 'c) vec" assumes "is_bezout_ext bezout" shows "det A = (1::'b) div fst (echelon_form_of_det A bezout) * det (snd (echelon_form_of_det A bezout))"
preprm_apply_id: fixes x :: "'b" shows "preprm_apply preprm_id x = x"
subst_pair_compose: fixes \<delta> :: "'e \<Rightarrow> ('c, 'f) Term.term"   and \<I> :: "'f \<Rightarrow> ('c, 'd) Term.term"   and d :: "('c, 'e) Term.term \<times> ('c, 'e) Term.term" shows "d \<cdot>\<^sub>p \<delta> \<circ>\<^sub>s \<I> = d \<cdot>\<^sub>p \<delta> \<cdot>\<^sub>p \<I>"
finite_succeed_t: fixes i :: "nat" shows "finite (succeed i) = finite (succeed_t i)"
simplex_ns_sat: fixes cs :: "('i \<times> 'a ns_constraint) list"   and v :: "(nat, 'a) mapping" assumes "solve_exec_ns cs = Inr v" shows "\<langle>v\<rangle> \<Turnstile>\<^sub>n\<^sub>s\<^sub>s  (snd ` set cs)"
N2_Delta2_E1_disjoint: shows "N\<^bsub>\<V>2\<^esub> \<inter> \<Delta>\<^bsub>\<Gamma>2\<^esub> \<inter> E\<^bsub>ES1\<^esub> = {}"
functional_list_def: fixes xs :: "('c \<times> 'd) list" shows "functional_list xs = (\<forall>x y z. ListMem (x, y) xs \<and> ListMem (x, z) xs \<longrightarrow> y = z)"
timpls_transformable_to_pred_finite_aux1: fixes f :: "('b, 'c, 'd) prot_fun"   and A :: "'d set set"   and T :: "(('b, 'c, 'd) prot_fun, 'e) Term.term list" assumes "\<not> is_Abs f" shows "{s. timpls_transformable_to_pred A (Fun f T) s} \<subseteq> Fun f ` {S. length T = length S \<and> (\<forall>s\<in>set S. \<exists>t\<in>set T. timpls_transformable_to_pred A t s)}"
map_preserves_dom: fixes f :: "'c" assumes "Cop.arr f" shows "local.map (Cop.dom f) = S.dom (local.map f)"
simps: fixes ant :: "'a form multiset"   and suc :: "'a form multiset" shows "succ ( ant \<Rightarrow>* suc) = suc"
stable_def: fixes \<mu> :: "('m \<times> 'w) set" shows "stable \<mu> = (match \<mu> \<and> (\<forall>m w. \<not> blocks m w \<mu>))"
block_closed: fixes a :: "'c set \<times> 'd"   and A :: "('c set \<times> 'd) set" assumes "a \<in> A"   and "A \<in> eqY.Partition" shows "a \<in> Sigma {U. is_open\<^sub>Y U \<and> f x \<in> U} \<O>\<^sub>Y"
sec_type_sec_type': fixes l :: "nat"   and c :: "com" assumes "l \<turnstile> c" shows "l \<turnstile>' c"
PiE_iff: fixes f :: "'f \<Rightarrow> 'g"   and I :: "'f set"   and X :: "'f \<Rightarrow> 'g set" shows "(f \<in> Pi\<^sub>E I X) = ((\<forall>i\<in>I. f i \<in> X i) \<and> f \<in> extensional I)"
inject: fixes x21 :: "'f"   and x22 :: "'f list"   and y21 :: "'f"   and y22 :: "'f list" shows "(x21 # x22 = y21 # y22) = (x21 = y21 \<and> x22 = y22)"
right_associated_product: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "CM"   and "\<exists>w. P w x \<and> P w y \<and> P w z" shows "\<forall>w. P w (x \<otimes> (y \<otimes> z)) = (P w x \<and> P w y \<and> P w z)"
P_invariant_steps: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and a :: "'c"   and as :: "'c list" assumes "Graph_Invariant_Strong E P"   and "Graph_Defs.steps E (a # as)" shows "list_all P as"
SCallRedsFinal: fixes P :: "(char list list \<times> char list exp) prog"   and es :: "char list exp list"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and vs :: "val list"   and h\<^sub>2 :: "heap"   and l\<^sub>2 :: "sfields"   and sh\<^sub>2 :: "sheap"   and b\<^sub>2 :: "bool"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and pns :: "char list list"   and body :: "char list exp"   and D :: "char list"   and sfs :: "sfields"   and l\<^sub>2' :: "sfields"   and ef :: "char list exp"   and h\<^sub>3 :: "heap"   and l\<^sub>3 :: "sfields"   and sh\<^sub>3 :: "sheap"   and b\<^sub>3 :: "bool" assumes "wwf_J_prog P"   and "P \<turnstile> \<langle>es,s\<^sub>0,b\<^sub>0\<rangle> [\<rightarrow>]* \<langle>map Val vs,(h\<^sub>2, l\<^sub>2, sh\<^sub>2),b\<^sub>2\<rangle>"   and "P \<turnstile> C sees M, Static :  Ts\<rightarrow>T = (pns, body) in D"   and "sh\<^sub>2 D = \<lfloor>(sfs, Done)\<rfloor> \<or> M = clinit \<and> sh\<^sub>2 D = \<lfloor>(sfs, Processing)\<rfloor>"   and "length vs = length pns"   and "l\<^sub>2' = [pns [\<mapsto>] vs]"   and "P \<turnstile> \<langle>body,(h\<^sub>2, l\<^sub>2', sh\<^sub>2),False\<rangle> \<rightarrow>* \<langle>ef,(h\<^sub>3, l\<^sub>3, sh\<^sub>3),b\<^sub>3\<rangle>"   and "final ef" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sM(es),s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>ef,(h\<^sub>3, l\<^sub>2, sh\<^sub>3),b\<^sub>3\<rangle>"
last_zip: fixes xs :: "'c list"   and ys :: "'d list" assumes "xs \<noteq> []"   and "ys \<noteq> []"   and "length xs = length ys" shows "last (xs || ys) = (last xs, last ys)"
trancl_unfold: fixes r :: "('b \<times> 'b) set" shows "r\<^sup>+ = r \<union> r\<^sup>+ O r"
WEAK_ALL_list_def: fixes Xi :: "'b list"   and P_impl :: "'b \<Rightarrow> bool nres" shows "WEAK_ALL_list Xi P_impl \<equiv> WHILE (\<lambda>(xa, xb). xb \<and> xa \<noteq> []) (\<lambda>(xa, xb). RETURN (hd xa, tl xa) \<bind> (\<lambda>(xc, xd). P_impl xc \<bind> (\<lambda>xe. RETURN (xd, xe \<and> xb)))) (Xi, True) \<bind> (\<lambda>(x, y). RETURN y)"
intros_6: fixes xs1 :: "'b list"   and xs2 :: "'b list"   and xs3 :: "'b list" assumes "xs1 \<approx>2 xs2"   and "xs2 \<approx>2 xs3" shows "xs1 \<approx>2 xs3"
hyp_c2: shows "\<forall>s'. \<Gamma>\<turnstile> \<langle>c1,Normal s\<rangle> \<Rightarrow> Abrupt s' \<longrightarrow> \<Gamma>\<turnstile>c2 \<down> Normal s' \<and> \<not> AlternativeSmallStep.inf \<Gamma> [c2] [] (Normal s')"
combine_inv2_2: fixes lt :: "('a, 'b) rbt"   and rt :: "('a, 'b) rbt" assumes "inv2 lt"   and "inv2 rt"   and "bheight lt = bheight rt" shows "inv2 (combine lt rt)"
is_Done_def: fixes resumption :: "('f, 'g, 'h) Resumption.resumption" shows "Resumption.resumption.is_Done resumption = (\<exists>x1. resumption = Resumption.resumption.Done x1)"
cinner_mult_left': fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and cinner :: "'a \<Rightarrow> 'a \<Rightarrow> complex"   and a :: "'b"   and m :: "complex"   and b :: "'b" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner (a * of_complex m) b = cnj m * cinner a b"
pos_b: shows "0 < outer_measure_of M (K \<inter> {x \<in> space M. b n \<le> f x})"
ex_def: fixes P :: "'a \<Rightarrow> (nat \<Rightarrow> cnf) \<Rightarrow> nat \<Rightarrow> bool" shows "ex P \<equiv> \<lambda>t n. \<exists>y. P y t n"
weight_finite: fixes \<Gamma> :: "('a, 'b) web_scheme"   and x :: "'a" assumes "countable_bipartite_web \<Gamma>" shows "weight \<Gamma> x \<noteq> \<top>"
elims_2: fixes x :: "'a Matrix.mat" assumes "square_mat x" obtains A :: "'a Matrix.mat" where "x = A" and "dim_col A = dim_row A"
fact_1: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.semiring_char_0 one times plus zero" shows "semiring_char_0.fact one plus zero 1 = one"
succeed_alt_def: shows "succeed i = {x. \<exists>n. rank x n = Some i \<and> token_run x n = q\<^sub>0}"
proots_within_uminus: fixes p :: "'a poly"   and s :: "'a set" shows "proots_within (- p) s = proots_within p s"
homeomorphism_add: fixes c :: "'b" shows "homeomorphism UNIV UNIV (\<lambda>x. x + c) (\<lambda>x. x - c)"
fls_X_power_times_conv_shift_2: fixes f :: "'a fls"   and n :: "nat" shows "f * fls_X ^ n = fls_shift (- int n) f"
arr_RelI: fixes \<alpha> :: "V"   and T :: "V" assumes "\<Z> \<alpha>"   and "vfsequence T"   and "vbrelation (T\<lparr>ArrVal\<rparr>)"   and "vcard T = 3\<^sub>\<nat>"   and "\<D>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>) \<subseteq>\<^sub>\<circ> T\<lparr>ArrDom\<rparr>"   and "\<R>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>) \<subseteq>\<^sub>\<circ> T\<lparr>ArrCod\<rparr>"   and "T\<lparr>ArrDom\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"   and "T\<lparr>ArrCod\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>" shows "arr_Rel \<alpha> T"
cachedWeight_splitLeaf: fixes t :: "'a tree"   and w\<^sub>a :: "nat"   and a :: "'a"   and w\<^sub>b :: "nat"   and b :: "'a" shows "cachedWeight (splitLeaf t w\<^sub>a a w\<^sub>b b) = cachedWeight t"
elims_2: fixes x :: "'b heap" assumes "is_root' x" obtains "x = heap.Empty"   | xa :: "'b" and hs :: "'b heap list" where "x = Hp xa hs" and "no_Emptys hs"
least: fixes A :: "'a measure set"   and x :: "'a measure" assumes "\<And>z. z \<in> A \<Longrightarrow> z \<le> x" shows "Sup A \<le> x"
alg_dfs_def: fixes aOps :: "('c, 'b, 'e list) MapOps"   and tOps :: "('d, 'b \<times> 'f, 'b) MapOps"   and simObs :: "'b \<Rightarrow> 'f"   and simTrans :: "'b \<Rightarrow> 'b list"   and simAction :: "'b \<Rightarrow> 'e list" shows "alg_dfs aOps tOps simObs simTrans simAction \<equiv> let k_empt = \<lparr>aActs = MapOps.empty aOps, aTrans = MapOps.empty tOps\<rparr>; k_memb = \<lambda>s A. isSome (lookup aOps (aActs A) s); k_succs = simTrans; actsUpdate = \<lambda>ec A. update aOps ec (simAction ec) (aActs A); transUpdate = \<lambda>ec ec'. update tOps (ec, simObs ec') ec'; k_ins = \<lambda>ec A. \<lparr>aActs = actsUpdate ec A, aTrans = foldr (transUpdate ec) (k_succs ec) (aTrans A)\<rparr> in gen_dfs k_succs k_ins k_memb k_empt"
word_size: fixes w :: "'a Word.word" shows "size w = LENGTH('a)"
inverts_components: fixes a :: "'a \<times> 'a \<times> 'a" assumes "VVV.ide a" shows "inverse_arrows \<a>[fst a, fst (snd a), snd (snd a)] (\<alpha>' a)"
cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_inject: fixes x :: "(_) shadow_root_ptr"   and y :: "(_) shadow_root_ptr" shows "(cast x = cast y) = (x = y)"
simps_2: fixes ts :: "'a \<Rightarrow> ('b \<times> 'c \<Rightarrow>f nat) option"   and ta :: "('a, 'b, 'd) new_thread_action"   and tas :: "('a, 'b, 'd) new_thread_action list" shows "redT_updTs ts (ta # tas) = redT_updTs (redT_updT ts ta) tas"
borel_measurable_diff': fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "f - g \<in> borel_measurable M"
e_2: shows "{x \<in> space M. ennreal e \<le> f x} \<notin> null_sets M"
set_rec: fixes xs :: "'g list" shows "set xs = rec_list {} (\<lambda>x _. insert x) xs"
Limsup_ereal_mult_right: fixes F :: "'a filter"   and c :: "real"   and f :: "'a \<Rightarrow> ereal" assumes "F \<noteq> bot"   and "0 \<le> c" shows "Limsup F (\<lambda>n. f n * ereal c) = Limsup F f * ereal c"
h: shows "u = u1 @ [tick] \<and> t = t1 @ [tick] \<and> s setinterleaves ((t, u), insert tick (ev ` A))"
reachable_nodes_refl: fixes r :: "'a"   and g :: "'a ugraph" shows "r \<in> reachable_nodes g r"
starset_subset: fixes A :: "'c set"   and B :: "'c set" shows "(*s* A \<subseteq> *s* B) = (A \<subseteq> B)"
cases: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g list" assumes "class.linorder less_eq less" obtains "x = []"   | xa :: "'g" and ys :: "'g list" where "x = xa # ys"
infinite_arith_prog: fixes b :: "nat"   and a :: "int" assumes "0 < b" shows "infinite (arith_prog a b)"
subset_insert: fixes x :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "(x::'b) \<notin> (A::'b set)" shows "((A::'b set) \<subseteq> (x::'b) \<triangleright> (B::'b set)) = (A \<subseteq> B)"
R14_3: shows "pmf (sds R14) c = 1 - pmf (sds R14) a"
min_list_Cons: fixes x :: "'c"   and y :: "'c"   and xs :: "'c list"   and ys :: "'c list" assumes "x \<le> y"   and "length xs = length ys"   and "Missing_List.min_list xs \<le> Missing_List.min_list ys" shows "Missing_List.min_list (x # xs) \<le> Missing_List.min_list (y # ys)"
add_left_cancel: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and a :: "'k"   and b :: "'k"   and c :: "'k" assumes "class.cancel_semigroup_add plus" shows "(plus a b = plus a c) = (b = c)"
bot_dres_def: shows "dSUCCEED \<equiv> dSUCCEEDi"
genideal_ideal: fixes S :: "int set" assumes "S \<subseteq> UNIV" shows "ideal (Idl\<^bsub>\<Z>\<^esub> S) \<Z>"
OV_inv2_inductive_2: shows "{OV_inv2} TS.trans obsv_TS {> OV_inv2}"
gen_iapath_def: fixes V :: "'a set"   and u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" shows "S.gen_iapath V u p v \<equiv> u \<in> V \<and> v \<in> V \<and> S.apath u p v \<and> set (S.inner_verts p) \<inter> V = {} \<and> p \<noteq> []"
int_and_le: fixes a :: "int"   and y :: "int" assumes "bin_sign a = 0" shows "y AND a \<le> a"
that: fixes i :: "nat" assumes "is_kth_obs (path \<sigma>) k i" shows "thesis"
simps: fixes a :: "nat"   and x :: "nat"   and xs :: "nat list"   and s :: "nat" shows "incs a x (xs, s) = (let t = s + a * x in if cond (x # xs) t then (x # xs, t) # incs a (Suc x) (xs, s) else [])"
discrete_compactin_eq_finite: fixes S :: "'b set"   and X :: "'b topology" assumes "S \<inter> X derived_set_of S = {}" shows "compactin X S = (S \<subseteq> topspace X \<and> finite S)"
prod_mset_fract_poly: fixes f :: "'b \<Rightarrow> 'a poly"   and A :: "'b multiset" shows "(\<Prod>x\<in>#A. map_poly to_fract (f x)) = map_poly to_fract (\<Prod>\<^sub># (image_mset f A))"
wtE_pickE: fixes xl :: "Preliminaries.var list"   and al :: "univ list" assumes "length xl = length al"   and "distinct xl"   and "\<And>i. i < length xl \<Longrightarrow> eintT (GE.tpOfV (xl ! i)) (al ! i)" shows "wtE (pickE xl al)"
a2: shows "a + b * (1 - a\<^sup>2 - b\<^sup>2) = 0"
stranspose_eq_scons: fixes ws :: "'b stream list"   and a :: "'b list"   and w :: "'b list stream" shows "(stranspose ws = a ## w) = (map shd ws = a \<and> stranspose (map stl ws) = w)"
distinct_2: fixes x21 :: "'a"   and x22 :: "'a FOL_Fitting.term list"   and x1 :: "nat" shows "App x21 x22 \<noteq> Var x1"
term_pp_rel_sym: fixes r :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and a :: "'t \<times> 'a"   and b :: "'t \<times> 'a" assumes "symp r"   and "term_pp_rel r a b" shows "term_pp_rel r b a"
dist_fs_triangle_infinite2: fixes a :: "complex"   and c :: "complex" shows "1 / sqrt (1 + (cmod a)\<^sup>2) \<le> cmod (a - c) / (sqrt (1 + (cmod a)\<^sup>2) * sqrt (1 + (cmod c)\<^sup>2)) + 1 / sqrt (1 + (cmod c)\<^sup>2)"
i_less_tW: shows "Suc i \<le> length (takeWhile (\<lambda>ys. Suc j \<le> length ys) xs)"
Re_sgn: fixes R :: "complex" assumes "is_real R" shows "Re (sgn R) = sgn (Re R)"
inj_def: fixes f :: "'g \<Rightarrow> 'h" shows "inj f = (\<forall>x y. f x = f y \<longrightarrow> x = y)"
top_on_Step: fixes X :: "char list set"   and f :: "char list \<Rightarrow> aexp \<Rightarrow> 'b st option \<Rightarrow> 'b st option"   and g :: "bexp \<Rightarrow> 'b st option \<Rightarrow> 'b st option"   and C :: "'b st option acom"   and S :: "'b st option" assumes "\<And>x e S. \<lbrakk>top_on\<^sub>o S X; x \<notin> X; vars e \<subseteq> - X\<rbrakk> \<Longrightarrow> top_on\<^sub>o (f x e S) X"   and "\<And>b S. \<lbrakk>top_on\<^sub>o S X; vars b \<subseteq> - X\<rbrakk> \<Longrightarrow> top_on\<^sub>o (g b S) X"   and "vars C \<subseteq> - X"   and "top_on\<^sub>o S X"   and "top_on\<^sub>c C X" shows "top_on\<^sub>c (Step f g S C) X"
poly_of_pm_monomial: fixes x :: "'b"   and t :: "'b \<Rightarrow>\<^sub>0 nat"   and c :: "'a" shows "poly_of_pm x (monomial c t) = (Polynomial.monom c (lookup t x) when t \<in> .[{x}])"
IH_4: fixes t :: "real" assumes "0 < t" shows "nth_derivative k f t 1 = pk t / t ^ (2 * k) * exp (- inverse t)"
combine_altdef: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and m1 :: "('a, 'b) mapping"   and m2 :: "('a, 'b) mapping" shows "Mapping.combine f m1 m2 = Mapping.combine_with_key (\<lambda>_. f) m1 m2"
lexord_irreflexive: fixes r :: "('b \<times> 'b) set"   and xs :: "'b list" assumes "\<forall>x. (x, x) \<notin> r" shows "(xs, xs) \<notin> lexord r"
red_nonzero: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "red F p q" shows "p \<noteq> 0"
sign_simps_5: fixes ah :: "'d"   and bh :: "'d" shows "ah * bh = bh * ah"
minus_le_tminus: fixes a :: "real"   and b :: "real" shows "a - b \<le> a \<ominus> b"
pdevs_eqI: fixes x :: "'d pdevs"   and y :: "'d pdevs" assumes "\<And>i. pdevs_apply x i = pdevs_apply y i" shows "x = y"
insert_Diff_if: fixes x :: "'q"   and A :: "'q set"   and B :: "'q set" shows "insert x A - B = (if x \<in> B then A - B else insert x (A - B))"
valid_wp: fixes P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and p :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "(\<Turnstile>\<^sub>1 {P} c { p \<Down>Q}) = (\<exists>k>0. \<forall>l s. P l s \<longrightarrow> wp\<^sub>1 c Q l s \<and> \<down>\<^sub>t (c, s) \<le> k * p s)"
ineq: fixes k :: "nat" assumes "0 < k" shows "\<K> \<le> root k (real k) * root k (real max_len)"
rename_gens_normalize: fixes f :: "bool \<Rightarrow> bool"   and g :: "'c \<Rightarrow> 'd"   and l :: "(bool \<times> 'c) list" assumes "inj f"   and "inj_on g (snd ` set l)" shows "Cancelation.normalize (map (map_prod f g) l) = map (map_prod f g) (Cancelation.normalize l)"
insert_mono: fixes C :: "'e set"   and D :: "'e set"   and a :: "'e" assumes "C \<subseteq> D" shows "insert a C \<subseteq> insert a D"
minimal: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.pcpo below" shows "below (pcpo.bottom below) x"
mset_eq_length: fixes xs :: "'a list"   and ys :: "'a list" assumes "mset xs = mset ys" shows "length xs = length ys"
that: fixes asx :: "'edge list"   and asx' :: "'edge list"   and ms'' :: "'node list"   and s'' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "as = asx @ [last asx']"   and "S,f \<turnstile> (ms,s) =asx\<Rightarrow>* (ms'',s'')"   and "S,f \<turnstile> (ms'',s'') =asx'\<Rightarrow> (ms',s')" shows "thesis"
Unifier_in_unifiers_singleton: fixes s :: "('a, 'b) Term.term"   and \<theta> :: "'b \<Rightarrow> ('a, 'b) Term.term"   and t :: "('a, 'b) Term.term" shows "Unifier \<theta> s t = (\<theta> \<in> unifiers {(s, t)})"
assms: shows "(y, x) \<in> rtrancl_on F (r\<inverse>)"
collapse: fixes id :: "'d Monomorphic_Monad.id" shows "return_id (extract id) = id"
disc_2: fixes x21 :: "'a"   and x22 :: "'a llist" shows "\<not> lnull (x21 #\<^sub>l x22)"
widen_refl: fixes 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 T :: "ty" shows "P \<turnstile> T \<le> T"
\<tau>external_def: fixes P :: "'a prog"   and hT :: "htype"   and M :: "String.literal" shows "\<tau>external P hT M = (\<exists>Ts Tr D. P \<turnstile> class_type_of hT sees M: Ts\<rightarrow>Tr = Native in D \<and> \<tau>external_defs D M)"
stable_pair_on_stable_on: fixes ds :: "'b set"   and XD_XH :: "'a set \<times> 'a set" assumes "stable_pair_on ds XD_XH" shows "stable_on ds (match XD_XH)"
compact_diff: fixes "open" :: "'b set \<Rightarrow> bool"   and S :: "'b set"   and T :: "'b set" assumes "class.topological_space open"   and "topological_space.compact open S"   and "open T" shows "topological_space.compact open (S - T)"
1: shows "wf\<^sub>l\<^sub>s\<^sub>t X (c2 @ A)"
ordering_axioms: shows "ordering (le_of_nat_term_order cmp_term) (lt_of_nat_term_order cmp_term)"
simps_2: fixes x :: "'c"   and xs :: "'c list"   and y :: "'c" shows "index (x # xs) y = (if x = y then if x \<in> set xs then index xs y + 1 else 0 else index xs y + 1)"
vreal_ls_def: shows "vreal_ls = ZFC_in_HOL.set {[a, b]\<^sub>\<circ> |a b. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> \<real>\<^sub>\<circ> ^\<^sub>\<times> 2\<^sub>\<nat> \<and> real_of_vreal a < real_of_vreal b}"
dual_ccpo_mono_simp: fixes f :: "'g \<Rightarrow> 'h" shows "monotone (\<lambda>x y. y \<le> x) (\<lambda>x y. y \<le> x) f = mono f"
sqninv: fixes dests :: "nat \<Rightarrow> nat option"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and ip :: "nat"   and rsn :: "nat" assumes "\<forall>ip\<in>dom dests. ip \<in> kD rt \<and> sqn rt ip \<le> the (dests ip)"   and "dests ip = Some rsn" shows "sqn (invalidate rt dests) ip \<le> rsn"
simps_2: fixes p :: "'b \<Rightarrow> bool" shows "atoms_pltl atom\<^sub>p(p) = {p}"
reachable_def: fixes xs :: "'a set" shows "reachable xs \<equiv> {(x, y). y \<in> set (succs x)}\<^sup>* `` xs"
\<eta>_def: shows "\<eta> \<equiv> transformation_by_components.map (\<cdot>\<^sub>D) (\<cdot>\<^sub>D) D.map \<eta>o"
Suc_3: shows "DIM('a) + 1 \<le> Suc n"
vector_space_on_def: fixes U\<^sub>V\<^sub>S :: "'d set"   and scale :: "'e \<Rightarrow> 'd \<Rightarrow> 'd" shows "vector_space_on U\<^sub>V\<^sub>S scale \<equiv> module_on U\<^sub>V\<^sub>S scale"
pascal_prop_def: shows "pascal_prop \<equiv> \<forall>A B C D E F. is_pascal A B C D E F \<longrightarrow> is_pascal B A C D E F"
inv_plus': fixes a :: "'av"   and a1 :: "'av"   and a2 :: "'av"   and a\<^sub>1' :: "'av"   and a\<^sub>2' :: "'av"   and i1 :: "int"   and i2 :: "int" assumes "inv_plus' a a1 a2 = (a\<^sub>1', a\<^sub>2')"   and "i1 \<in> \<gamma> a1"   and "i2 \<in> \<gamma> a2"   and "i1 + i2 \<in> \<gamma> a" shows "i1 \<in> \<gamma> a\<^sub>1' \<and> i2 \<in> \<gamma> a\<^sub>2'"
rolling_rule: fixes g :: "'d \<rightarrow> 'c"   and f :: "'c \<rightarrow> 'd" shows "fix\<cdot>(g oo f) = g\<cdot>(fix\<cdot>(f oo g))"
rel_gpv''_Done: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and C :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and x :: "'d"   and y :: "'e" shows "rel_gpv'' A C R (Generative_Probabilistic_Value.Done x) (Generative_Probabilistic_Value.Done y) = A x y"
sqrt_aux_1: fixes n :: "nat" shows "finite {m. m\<^sup>2 \<le> n}"
solve_equation: fixes H :: "'a set"   and x :: "'a"   and y :: "'a" assumes "subgroup H G"   and "x \<in> H"   and "y \<in> H" shows "\<exists>h\<in>H. y = h \<otimes> x"
C_unfoldl: fixes x :: "'a" shows "(1::'a) + x \<cdot> x\<^sup>\<star> = x\<^sup>\<star>"
Hermite_reduce_above_Least: fixes A :: "(('a, 'b) vec, 'c) vec"   and ass :: "'a \<Rightarrow> 'a"   and i :: "'c"   and ia :: "'c"   and res :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "echelon_form A"   and "ass_function ass"   and "\<not> is_zero_row i A"   and "\<not> is_zero_row ia A"   and "\<not> is_zero_row ia (Hermite_reduce_above (mult_row A i (ass (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a))) div A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a)))) (mod_type_class.to_nat i) i (LEAST n. A $ i $ n \<noteq> (0::'a)) res)" shows "(LEAST n. Hermite_reduce_above (mult_row A i (ass (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a))) div A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a)))) (mod_type_class.to_nat i) i (LEAST n. A $ i $ n \<noteq> (0::'a)) res $ ia $ n \<noteq> (0::'a)) = (LEAST n. A $ ia $ n \<noteq> (0::'a))"
cases: fixes a :: "'node SDG_node"   and S :: "'node SDG_node set" assumes "a \<in> combine_SDG_slices S" obtains n :: "'node SDG_node" where "a = n__" and "n__ \<in> S"   | n' :: "'node SDG_node" and n'' :: "'node SDG_node" and p :: "'pname" and n :: "'node SDG_node" where "a = n" and "n' \<in> S" and "n'' s-p\<rightarrow>\<^bsub>ret\<^esub> CFG_node (parent_node n')" and "n \<in> sum_SDG_slice2 n'"
hyps: assumes "a \<in> all_acquired xs" shows "(\<exists>sop a' v ys zs A L R W. xs = ys @ Write\<^sub>s\<^sub>b True a' sop v A L R W # zs \<and> a \<in> A) \<or> (\<exists>A L R W ys zs. xs = ys @ Ghost\<^sub>s\<^sub>b A L R W # zs \<and> a \<in> A)"