hcju/isabelleps huggingface dataset:
F2_antimono: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and ds :: "'e set" assumes "ContractsWithSubstitutes Xd Xh Pd Ch" shows "antimono (Contracts.F2 Pd ds)"
small_arr_Par: fixes \<alpha> :: "V" shows "small {T. arr_Par \<alpha> T}"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (And p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Or p q)"   and "\<And>c e. P (Eq (CN 0 c e))"   and "\<And>c e. P (NEq (CN 0 c e))"   and "\<And>c e. P (Lt (CN 0 c e))"   and "\<And>c e. P (Le (CN 0 c e))"   and "\<And>c e. P (Gt (CN 0 c e))"   and "\<And>c e. P (Ge (CN 0 c e))"   and "P T"   and "P F"   and "\<And>va. P (Lt (C va))"   and "\<And>va. P (Lt (Bound va))"   and "\<And>vd vb vc. P (Lt (CN (Suc vd) vb vc))"   and "\<And>va. P (Lt (Neg va))"   and "\<And>va vb. P (Lt (Add va vb))"   and "\<And>va vb. P (Lt (Sub va vb))"   and "\<And>va vb. P (Lt (Mul va vb))"   and "\<And>va. P (Le (C va))"   and "\<And>va. P (Le (Bound va))"   and "\<And>vd vb vc. P (Le (CN (Suc vd) vb vc))"   and "\<And>va. P (Le (Neg va))"   and "\<And>va vb. P (Le (Add va vb))"   and "\<And>va vb. P (Le (Sub va vb))"   and "\<And>va vb. P (Le (Mul va vb))"   and "\<And>va. P (Gt (C va))"   and "\<And>va. P (Gt (Bound va))"   and "\<And>vd vb vc. P (Gt (CN (Suc vd) vb vc))"   and "\<And>va. P (Gt (Neg va))"   and "\<And>va vb. P (Gt (Add va vb))"   and "\<And>va vb. P (Gt (Sub va vb))"   and "\<And>va vb. P (Gt (Mul va vb))"   and "\<And>va. P (Ge (C va))"   and "\<And>va. P (Ge (Bound va))"   and "\<And>vd vb vc. P (Ge (CN (Suc vd) vb vc))"   and "\<And>va. P (Ge (Neg va))"   and "\<And>va vb. P (Ge (Add va vb))"   and "\<And>va vb. P (Ge (Sub va vb))"   and "\<And>va vb. P (Ge (Mul va vb))"   and "\<And>va. P (Eq (C va))"   and "\<And>va. P (Eq (Bound va))"   and "\<And>vd vb vc. P (Eq (CN (Suc vd) vb vc))"   and "\<And>va. P (Eq (Neg va))"   and "\<And>va vb. P (Eq (Add va vb))"   and "\<And>va vb. P (Eq (Sub va vb))"   and "\<And>va vb. P (Eq (Mul va vb))"   and "\<And>va. P (NEq (C va))"   and "\<And>va. P (NEq (Bound va))"   and "\<And>vd vb vc. P (NEq (CN (Suc vd) vb vc))"   and "\<And>va. P (NEq (Neg va))"   and "\<And>va vb. P (NEq (Add va vb))"   and "\<And>va vb. P (NEq (Sub va vb))"   and "\<And>va vb. P (NEq (Mul va vb))"   and "\<And>v va. P (Dvd v va)"   and "\<And>v va. P (NDvd v va)"   and "\<And>v. P (NOT v)"   and "\<And>v va. P (Imp v va)"   and "\<And>v va. P (Iff v va)"   and "\<And>v. P (E v)"   and "\<And>v. P (A v)"   and "\<And>v. P (Closed v)"   and "\<And>v. P (NClosed v)" shows "P a0"
f1: shows " t1 \<longrightarrow>\<^sub>1\<^sup>* t3"
im_sheaf_axioms: shows "Comm_Ring.im_sheaf S is_open \<FF> \<rho> b add_str mult_str zero_str one_str S is_open (identity S)"
isomorphic_objects_are_equal: fixes a :: "'a"   and b :: "'a" assumes "obj a"   and "obj b"   and "a \<cong> b" shows "a = b"
phull_Macaulay_list: fixes ps :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "phull (set (Macaulay_list ps)) = phull (set ps)"
cod_simps_2: shows "\<mu>\<nu>_\<pi>.cod.leg0 = \<mu>\<nu>_\<pi>.leg0"
sup_left_unit: fixes x :: "'a" shows "bot \<squnion> - x = - x"
TokensAt_subset_\<X>: fixes k :: "nat"   and I :: "item set" shows "TokensAt k I \<subseteq> \<X> k"
t4_space: fixes "open" :: "'b set \<Rightarrow> bool"   and S :: "'b set"   and T :: "'b set" assumes "class.t4_space open"   and "topological_space.closed open S"   and "topological_space.closed open T"   and "S \<inter> T = {}" shows "\<exists>U V. open U \<and> open V \<and> S \<subseteq> U \<and> T \<subseteq> V \<and> U \<inter> V = {}"
rbd_set: fixes R :: "('d \<times> 'c) set"   and X :: "'d set" shows "bd\<^sub>\<R> R X = {y. \<exists>x\<in>X. s2p R (x, y)}"
minus_Set: fixes A :: "'b set"   and t :: "('b, unit) RBT.rbt" shows "A - Set t = fold_keys Set.remove t A"
delete_edge_valid: fixes g :: "('c, 'd) graph"   and v :: "'c"   and e :: "'d"   and v' :: "'c" assumes "valid_graph g" shows "valid_graph (delete_edge v e v' g)"
X_def: fixes M :: "'a measure"   and I :: "'b set"   and A :: "'b \<Rightarrow> 'a set"   and i :: "'b" assumes "prob_space_with_indicators M I A" shows "prob_space_with_indicators.X A i = rind (A i)"
red2_rtrancl_\<tau>_heapD: fixes t :: "'t"   and s2 :: "'x2 \<times> 'm2"   and s2' :: "'x2 \<times> 'm2"   and s1 :: "'x1 \<times> 'm1" assumes "r2.silent_moves t s2 s2'"   and "t \<turnstile> s1 \<approx> s2" shows "snd s2' = snd s2"
Comp: fixes G :: "prog"   and s0 :: "st"   and c1 :: "stmt"   and s1 :: "state"   and c2 :: "stmt"   and s2 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>c1\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>c2\<rightarrow> s2" shows "G\<turnstile>Norm s0 \<midarrow>c1;; c2\<rightarrow> s2"
unat_of_nat_less: fixes a :: "'a word"   and b :: "'a word"   and c :: "nat" assumes "a < b"   and "unat b = c" shows "a < word_of_nat c"
I: fixes i :: "'a" assumes "i \<in> I" shows "A i \<in> fmeasurable M"
vcomp_transformation_cone: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b"   and a :: "'b"   and \<chi> :: "'a \<Rightarrow> 'b"   and D' :: "'a \<Rightarrow> 'b"   and \<tau> :: "'a \<Rightarrow> 'b" assumes "Limit.cone J C D a \<chi>"   and "natural_transformation J C D D' \<tau>" shows "Limit.cone J C D' a (vertical_composite.map J C \<chi> \<tau>)"
kad_def: fixes f :: "'c \<Rightarrow> 'd set" shows "kad f = (\<lambda>x. if f x = {} then \<eta> x else {})"
axioms_1: fixes G :: "('a, 'b) Graph_scheme" assumes "Tree G" shows "Graph G"
entailsD: fixes P :: "'a \<Rightarrow> real"   and Q :: "'a \<Rightarrow> real"   and s :: "'a" assumes "P \<tturnstile> Q" shows "P s \<le> Q s"
hypnat_diff_diff_left: fixes i :: "nat star"   and j :: "nat star"   and k :: "nat star" shows "i - j - k = i - (j + k)"
inf_set_def: fixes A :: "'e set"   and B :: "'e set" shows "A \<inter> B = Collect (inf (\<lambda>x. x \<in> A) (\<lambda>x. x \<in> B))"
ACI_norm_alt: fixes r :: "'b rexp" shows "ACI_norm_alt r = \<guillemotleft>r\<guillemotright>"
arc_conv_closed: fixes x :: "'a" assumes "bijective (x * top) \<and> bijective (x[[top]])" shows "bijective (x[[top]]) \<and> bijective (x\<^sup>T[[top]])"
ResB: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and y :: "name" assumes "P \<Longrightarrow>\<^sub>l\<^sup>^a<\<nu>x> \<prec> P'"   and "y \<noteq> a"   and "y \<noteq> x"   and "x \<sharp> P" shows "<\<nu>y>P \<Longrightarrow>\<^sub>l\<^sup>^a<\<nu>x> \<prec> <\<nu>y>P'"
assms: shows "\<forall>k<n. \<parallel>cid\<parallel>\<^bsub>t k\<^esub>"
nprv_impI_rev: fixes F :: "'fmla set"   and \<phi> :: "'fmla"   and \<psi> :: "'fmla" assumes "nprv F (imp \<phi> \<psi>)"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla"   and "\<psi> \<in> fmla" shows "nprv (insert \<phi> F) \<psi>"
subspace_tangent_space: fixes charts :: "('c, 'd) chart set"   and k :: "enat"   and p :: "'c" assumes "c_manifold charts k" shows "subspace (c_manifold.tangent_space charts k p)"
simps_7: fixes wf_atom :: "nat \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "nat"   and r :: "'b rexp" shows "alphabet.wf wf_atom n (Star r) = alphabet.wf wf_atom n r"
condD_4: fixes W :: "'a vec set" assumes "cond W"   and "X \<subseteq> \<int>\<^sub>v \<inter> Bounded_vec Bnd" shows "normal_vector W \<in> \<int>\<^sub>v \<inter> Bounded_vec (det_bound (n - 1) Bnd)"
less_fun_def: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "less_fun f g = (\<exists>k. f k < g k \<and> (\<forall>k'<k. f k' = g k'))"
maxsimp_connect: fixes x :: "'a set"   and y :: "'a set" assumes "x \<noteq> y"   and "domain.chamber x"   and "domain.chamber y" shows "\<exists>xs. domain.gallery (x # xs @ [y])"
map_consI_1: fixes w :: "'a list"   and f :: "'b \<Rightarrow> 'a"   and ww :: "'b list"   and a :: "'b" assumes "w = map f ww" shows "f a # w = map f (a # ww)"
group_part_aux_induct: fixes P :: "'b \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> 'a \<Rightarrow> 'a list \<Rightarrow> bool"   and a0a :: "'b"   and a1a :: "'a list"   and a2a :: "'a list"   and a2b :: "'a"   and a2c :: "'a list" assumes "\<And>p ls eq gs. P p [] ls eq gs"   and "\<And>x p xs ls eq gs. \<lbrakk>f x < p; P p xs (x # ls) eq gs\<rbrakk> \<Longrightarrow> P p (x # xs) ls eq gs"   and "\<And>x p xs ls eq gs. \<lbrakk>p < f x; P p xs ls eq (x # gs)\<rbrakk> \<Longrightarrow> P p (x # xs) ls eq gs"   and "\<And>x p xs ls eqa gs. \<lbrakk>f x = p; P p xs ls (merge x eqa) gs\<rbrakk> \<Longrightarrow> P p (x # xs) ls eqa gs" shows "P a0a a1a a2a a2b a2c"
hom_sum: fixes f :: "'d \<Rightarrow> 'c poly"   and X :: "'d set" shows "poly_lift (sum f X) = (\<Sum>x\<in>X. poly_lift (f x))"
h': shows "h3 \<turnstile> set_disconnected_nodes document_ptr (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr # disc_nodes_h3) \<rightarrow>\<^sub>h h'"
enum_all_fun_def: fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> bool" shows "enum_all P = all_n_lists (\<lambda>bs. P (the \<circ> map_of (zip enum bs))) (length enum)"
gfp_lfp': fixes f :: "'b set \<Rightarrow> 'b set" shows "- gfp f = lfp (\<lambda>s. - f (- s))"
lookup_le_poly_mapping_size: fixes g :: "'b \<Rightarrow> nat"   and m :: "'c \<Rightarrow>\<^sub>0 'b"   and k :: "'c"   and f :: "'c \<Rightarrow> nat" shows "g (lookup m k) \<le> poly_mapping_size f g m"
finite_has_minimal: fixes A :: "'a set" assumes "finite A"   and "A \<noteq> {}" shows "\<exists>m\<in>A. \<forall>b\<in>A. b \<le> m \<longrightarrow> m = b"
cbig_def: fixes small :: "'a \<Rightarrow> 'b \<Rightarrow> 'b set"   and endset :: "'b set"   and combine :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and collect_id :: "'c"   and collect :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c"   and P :: "'a"   and \<sigma> :: "'b" assumes "CollectionSemantics small endset combine collect_id" shows "CollectionSemantics.cbig small endset collect combine collect_id P \<sigma> \<equiv> {(\<sigma>', coll). \<exists>n. (\<sigma>', coll) \<in> CollectionSemantics.csmall_nstep small collect combine collect_id P \<sigma> n \<and> \<sigma>' \<in> endset}"
heap_heap_of_A: fixes xs :: "'b list" shows "heap (heap_of_A xs)"
Cauchy_theorem_flat_lemma: fixes a :: "complex"   and b :: "complex"   and c :: "complex"   and f :: "complex \<Rightarrow> complex"   and k :: "real" assumes "continuous_on (convex hull {a, b, c}) f"   and "c - a = k *\<^sub>R (b - a)"   and "0 \<le> k" shows "contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f = 0"
semIntAll_termFSwSbImorph: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'a) semDom" assumes "wlsSEM SEM" shows "termFSwSbImorph (semInt SEM) (semIntAbs SEM) (asIMOD SEM)"
m: shows "m \<equiv> LEAST m. k < m \<and> m \<le> n + r \<and> (\<pi> @\<^bsup>n\<^esup> \<pi>') m = ipd (\<pi> k)"
case_eq_if: fixes f1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and f3 :: "'b \<Rightarrow> 'd \<Rightarrow> 'c \<Rightarrow> 'a"   and f4 :: "'b \<Rightarrow> 'a"   and pac_step :: "('c, 'd, 'b) pac_step" shows "(case pac_step of Add x xa xb xc \<Rightarrow> f1 x xa xb xc | Mult x xa xb xc \<Rightarrow> f2 x xa xb xc | Extension x xa xb \<Rightarrow> f3 x xa xb | Del x \<Rightarrow> f4 x) = (if is_Add pac_step then f1 (pac_src1 pac_step) (pac_src2 pac_step) (new_id pac_step) (pac_res pac_step) else if is_Mult pac_step then f2 (pac_src1 pac_step) (pac_mult pac_step) (new_id pac_step) (pac_res pac_step) else if is_Extension pac_step then f3 (new_id pac_step) (new_var pac_step) (pac_res pac_step) else f4 (pac_src1 pac_step))"
l1x: shows "{y. real_of_1 (p, l, r) \<le> y \<and> y \<le> real_of_rat r \<and> poly (real_of_int_poly p) y = 0} = {real_of_1 (p, l, r)}"
closedin_subset_trans: fixes U :: "'c set"   and S :: "'c set"   and T :: "'c set" assumes "closedin (top_of_set U) S"   and "S \<subseteq> T"   and "T \<subseteq> U" shows "closedin (top_of_set T) S"
assms: fixes i :: "V" assumes "i \<in>\<^sub>\<circ> I" shows "f i \<in>\<^sub>\<circ> A i"
SUP_emeasure_incseq: fixes A :: "nat \<Rightarrow> 'b set"   and M :: "'b measure" assumes "range A \<subseteq> sets M"   and "incseq A" shows "(\<Squnion>n. emeasure M (A n)) = emeasure M (\<Union> (range A))"
Ostrowski_base_hom: fixes n :: "nat"   and vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant" assumes "vals_nonequiv K (Suc n) vv" shows "\<Omega>\<^bsub>K vv Suc n\<^esub> \<in> {h. h \<le> Suc n} \<rightarrow> carrier K"
zero_in_complex_cone: fixes a :: "real"   and b :: "real" assumes "a \<le> b" shows "0 \<in> complex_cone a b"
sel_2: fixes x1 :: "'b literal multiset multiset"   and x2 :: "'b literal multiset"   and x3 :: "'b literal multiset" shows "old_main_prem_of (old_Infer x1 x2 x3) = x2"
IClock: shows "\<forall>p t i. correct p t \<longrightarrow> IC p i t = PC p t + Adj p i"
Max_torder_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "Max_torder r = semilattice_set.F (max_torder r)"
trancl_id: fixes r :: "('d \<times> 'd) set" assumes "trans r" shows "r\<^sup>+ = r"
prop_atoms_notin_3: fixes \<phi>\<^sub>1 :: "'b ltln"   and \<phi>\<^sub>2 :: "'b ltln"   and \<phi> :: "'b ltln" shows "\<phi>\<^sub>1 and\<^sub>n \<phi>\<^sub>2 \<notin> prop_atoms \<phi>"
ereal_limsup_add_mono: fixes u :: "nat \<Rightarrow> ereal"   and v :: "nat \<Rightarrow> ereal" shows "limsup (\<lambda>n. u n + v n) \<le> limsup u + limsup v"
numeral_less_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and m :: "Num.num"   and n :: "Num.num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less (numeral.numeral one plus m) (numeral.numeral one plus n) = (m < n)"
ba_inf: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> - y = - (- - x \<squnion> - - y)"
simps: fixes base :: "'a word"   and len :: "nat"   and p_ip :: "'a word" shows "simple_match_ip (base, len) p_ip = (p_ip \<in> ipset_from_cidr base len)"
M: fixes m :: "nat"   and n :: "nat" assumes "M \<le> m"   and "M \<le> n" shows "C + deltaG TYPE('a) \<le> Gromov_product_at a (u m) (u n)"
mult_cancel_right: fixes f2 :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g2 :: "'a \<Rightarrow> 'b"   and f1 :: "'a \<Rightarrow> 'b"   and g1 :: "'a \<Rightarrow> 'b" assumes "f2 \<in> \<Theta>[F](g2)"   and "\<forall>\<^sub>F x in F. g2 x \<noteq> (0::'b)" shows "((\<lambda>x. f1 x * f2 x) \<in> O[F](\<lambda>x. g1 x * g2 x)) = (f1 \<in> O[F](g1))"
fun_add_eval_car: fixes R :: "('c, 'd) ring_scheme"   and a :: "'e"   and S :: "'e set"   and f :: "'e \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'c" assumes "ring_functions R"   and "a \<in> S"   and "f \<in> carrier (function_ring S R)"   and "g \<in> carrier (function_ring S R)" shows "(f \<oplus>\<^bsub>function_ring S R\<^esub> g) a = f a \<oplus>\<^bsub>R\<^esub> g a"
prems: shows "\<Gamma>\<turnstile> \<langle>Catch c1 c2,Normal s\<rangle> \<Rightarrow>\<notin>Fault ` F"
chop_def: fixes t :: "('c, 'd) tm" shows "chop t = apps (hd (args t)) (tl (args t))"
subdivision_comb_planar: fixes G :: "('c, 'd) pre_digraph"   and rev_G :: "'d \<Rightarrow> 'd"   and H :: "('c, 'd) pre_digraph"   and rev_H :: "'d \<Rightarrow> 'd" assumes "subdivision (G, rev_G) (H, rev_H)"   and "comb_planar H" shows "comb_planar G"
mono: fixes w :: "'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "w \<le> x"   and "y \<le> z" shows "f w y \<le> f x z"
phl_cons2: fixes w :: "'a"   and x :: "'a"   and z :: "'a"   and y :: "'a" assumes "w \<le> x"   and "z \<cdot> y \<le> y \<cdot> w" shows "z \<cdot> y \<le> y \<cdot> x"
cases: fixes a :: "'g"   and r :: "('h \<times> 'g) set" assumes "a \<in> Range r" obtains aa :: "'h" and b :: "'g" where "a = b" and "(aa, b) \<in> r"
Y: assumes "distincts yss"   and "map set xss = map set yss" shows "list_all2 (\<lambda>x ys. x \<in> set ys) yss (map permutations xss)"
RPCRelayArg_def: fixes ra :: "rpcOp" shows "RPCRelayArg ra \<equiv> case ra of memcall m \<Rightarrow> m"
ftf_syn32: fixes u :: "'c event list"   and u1 :: "'c event list"   and t :: "'c event list"   and t1 :: "'c event list"   and s :: "'c event list"   and A :: "'c set" assumes "u = u1 @ [tick] \<and> t = t1 @ [tick] \<and> s setinterleaves ((t, u), insert tick (ev ` A))" shows "\<exists>s1. s1 setinterleaves ((t1, u1), insert tick (ev ` A)) \<and> s = s1 @ [tick]"
shunt: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(ad x \<le> ad y + ad z) = (ad x \<cdot> d y \<le> ad z)"
distinct_1: fixes x2 :: "nat" shows "Server \<noteq> Friend x2"
cc_def: shows "cc = (\<lambda>i. (\<Sum>k = 0..<n. if k = i - (m - 1) then P $$ (j, k) else 0) * u $v i)"
dotprod_eq_nonzero_iff: fixes a :: "nat list"   and x :: "nat list"   and b :: "nat list"   and y :: "nat list" assumes "a \<bullet> x = b \<bullet> y"   and "length x = length a"   and "length y = length b"   and "0 \<notin> set a"   and "0 \<notin> set b" shows "nonzero x = nonzero y"
assoc: fixes a :: "'d"   and b :: "'d"   and c :: "'d" shows "max (max a b) c = max a (max b c)"
case_cong: fixes ename :: "ename"   and ename' :: "ename"   and f1 :: "vname \<Rightarrow> 'b"   and g1 :: "vname \<Rightarrow> 'b"   and f2 :: "'b"   and g2 :: "'b" assumes "ename = ename'"   and "\<And>x1. ename' = VNam x1 \<Longrightarrow> f1 x1 = g1 x1"   and "ename' = Res \<Longrightarrow> f2 = g2" shows "(case ename of VNam x1 \<Rightarrow> f1 x1 | Res \<Rightarrow> f2) = (case ename' of VNam x1 \<Rightarrow> g1 x1 | Res \<Rightarrow> g2)"
inject_2: fixes x3 :: "'b"   and y3 :: "'b" shows "(Inca.instr.ILoad x3 = Inca.instr.ILoad y3) = (x3 = y3)"
simps_15: fixes prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and l :: "floatarith"   and r :: "floatarith"   and env :: "taylor_model list" shows "approx_tm prec ord I a (Min l r) env = (case (approx_tm prec ord I a l env, approx_tm prec ord I a r env) of (None, b) \<Rightarrow> None | (Some t1, None) \<Rightarrow> None | (Some t1, Some t2) \<Rightarrow> Some (tm_min prec I a t1 t2))"
perm_swap_def: fixes x :: "'c"   and y :: "'c"   and f :: "'c \<Rightarrow> 'c" shows "perm_swap x y f \<equiv> x \<rightleftharpoons>\<^sub>F y \<circ> f \<circ> x \<rightleftharpoons>\<^sub>F y"
f_def: shows "f \<equiv> \<lambda>x. if x \<le> 0 then 0 else f0 x"
regular_limit: fixes f :: "'a set \<Rightarrow> 'a set" assumes "regular f" shows "regular (limit f)"
g': assumes "x \<in> T \<union> closure S \<inter> closure T" shows "(g has_derivative g' x) (at x within T \<union> closure S \<inter> closure T)"
simps_2: fixes x :: "'b"   and xs :: "'b list"   and v :: "nat" shows "(x # xs) !. v = (if length xs = v then Some x else xs !. v)"
range: shows "(ix p d + 1) * 2 ^ (lv p' d - lv p d) < ix p' d \<or> ix p' d < (ix p d - 1) * 2 ^ (lv p' d - lv p d)"
prodsums_eq_sumprods: fixes p :: "nat"   and m :: "nat"   and n :: "nat" assumes "coprime p m" shows "\<Sum> ((^) p ` {0..n}) * sigma m = \<Sum> {p ^ k * b |k b. k \<le> n \<and> b dvd m}"
stock_filtration: shows "G = stoch_proc_filt M geom_proc borel"
minus: fixes x :: "'a" assumes "vectorspace V"   and "x \<in> V" shows "\<parallel>- x\<parallel> = \<parallel>x\<parallel>"
size_pratt_fpc: fixes a :: "nat"   and p :: "nat"   and r :: "nat"   and qs :: "nat list" assumes "a \<le> p"   and "r \<le> p"   and "0 < a"   and "0 < r"   and "0 < p"   and "prod_list qs = r" shows "\<forall>x\<in>set (build_fpc p a r qs). Pratt_Certificate.size_pratt x \<le> 3 * log 2 (real p)"
Eats: fixes t1 :: "dbtm"   and t2 :: "dbtm" assumes "wf_dbtm t1"   and "wf_dbtm t2" shows "wf_dbtm (DBEats t1 t2)"
strict_trans2: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset" assumes "a \<subset># b"   and "b \<subseteq># c" shows "a \<subset># c"
assms_1: shows "a \<in>\<^sub>\<circ> r -`\<^sub>\<circ> A"
tendsto_at_right_sequentially: fixes a :: "'b"   and b :: "'b"   and X :: "'b \<Rightarrow> 'c"   and L :: "'c" assumes "a < b"   and "\<And>S. \<lbrakk>\<And>n. a < S n; \<And>n. S n < b; decseq S; S \<longlonglongrightarrow> a\<rbrakk> \<Longrightarrow> (\<lambda>n. X (S n)) \<longlonglongrightarrow> L" shows "(X \<longlongrightarrow> L) (at_right a)"
Diff_eq: fixes A :: "'c set"   and B :: "'c set" shows "A - B = A \<inter> - B"
alphaAbs_qAbs_imp_all_equal_or_qFresh: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGood X"   and "qAbs xs x X $= qAbs xs' x' X'" shows "xs = xs' \<and> (\<forall>y. (y = x \<or> qFresh xs y X) \<and> (y = x' \<or> qFresh xs y X') \<longrightarrow> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
in_cfls_in_cfl: fixes G :: "('a, 'b) cfg"   and P :: "('a + 'b) list fset"   and w :: "'a list" assumes "in_cfls G P w" shows "fBex P (\<lambda>\<alpha>. in_cfl G \<alpha> w)"
fps_XD'_def: fixes a :: "'b fps" shows "fps_XD' a = (\<lambda>b. a * fps_deriv b)"
False: shows "MPoly_Type.degree (p * q) var \<noteq> 0"
length_map2: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and xs :: "'b list"   and ys :: "'c list" shows "length (map2 f xs ys) = min (length xs) (length ys)"
Crypt_used_imp_spies: fixes K :: "nat"   and X :: "msg"   and evs :: "event list" assumes "Crypt K X \<in> used evs"   and "evs \<in> zg" shows "Crypt K X \<in> parts (knows Spy evs)"
2: shows "VVV.iso (\<lbrace>t1\<rbrace>, \<lbrace>t2\<rbrace>, \<lbrace>t3\<rbrace>)"
one_dimvec_def: shows "1 = dim_lambda (\<lambda>i. 0::'a)"
t_BIT_def: fixes n :: "nat" shows "t_BIT n = T_on_rand'_n BIT (fst BIT init \<bind> (\<lambda>is. return_pmf (init, is))) qs n"
alphas_abs_eqvt_1: fixes bs :: "atom set"   and x :: "'a"   and cs :: "atom set"   and y :: "'a"   and p :: "perm" assumes "(bs, x) \<approx>abs_set (cs, y)" shows "(p \<bullet> bs, p \<bullet> x) \<approx>abs_set (p \<bullet> cs, p \<bullet> y)"
finite_fold_rderiv: fixes \<phi> :: "('a, 'k) aformula" shows "finite {\<langle>fold rderiv xs \<langle>\<phi>\<rangle>\<rangle> |xs. True}"
mset_unplusm_dist_cases: fixes s :: "'a"   and A :: "'a multiset"   and B :: "'a multiset"   and C :: "'a multiset" assumes "add_mset s A = B + C" obtains "B = add_mset s (B - {#s#})" and "A = B - {#s#} + C"   | "C = add_mset s (C - {#s#})" and "A = B + (C - {#s#})"
interval_imp_mem: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x \<in> interval r a b" shows "(a, x) \<in> r"
not_eq_order_implies_strict: fixes a :: "'d"   and b :: "'d" assumes "a \<noteq> b"   and "a \<le> b" shows "a < b"
dom_dom: fixes f :: "'a \<times> 'a" assumes "VoV.arr f" shows "VoV.dom (VoV.dom f) = VoV.dom f"
cod_eqI': fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "a \<in> codomains f" shows "a = cod f"
alloc_def: fixes xs :: "'d list"   and h :: "heap" shows "Array.alloc xs h = (let l = lim h; r = Array l in Let (Array.set r xs (h\<lparr>lim := l + 1\<rparr>)) (Pair r))"
subgame_valid_path: fixes P :: "'a llist"   and V' :: "'a set" assumes "G'.valid_path P"   and "lset P \<subseteq> V'" shows "Digraph.valid_path (G'.subgame V') P"
map_is_TVarb: fixes \<Gamma>' :: "binding list"   and \<Gamma> :: "binding list"   and i :: "nat"   and T :: "POPLmark.type" assumes "map is_TVarB \<Gamma>' = map is_TVarB \<Gamma>"   and "\<Gamma>\<langle>i\<rangle> = \<lfloor>TVarB T\<rfloor>" shows "\<exists>T. \<Gamma>'\<langle>i\<rangle> = \<lfloor>TVarB T\<rfloor>"
hd_b_cons_in_nodes: shows "set (hd b # a ! hd b) \<subseteq> set nodes"
interaction_bound_IO_ignore: fixes out :: "'i"   and c :: "('h, 'i, 'j) rpv"   and gpv :: "('h, 'i, 'j) gpv"   and "consider" :: "'i \<Rightarrow> bool"   and input :: "'j" assumes "IO out c \<in> set_spmf (the_gpv gpv)"   and "\<not> consider out" shows "interaction_bound consider (c input) \<le> interaction_bound consider gpv"