hcju/isabelleps huggingface dataset:
xvecFreshPsi: shows "(p \<bullet> xvec) \<sharp>* \<Psi>"
idempotent: fixes f :: "'a upred \<Rightarrow> 'a upred"   and x :: "'a upred" assumes "Idem\<^bsub>\<P>\<^esub> f"   and "x \<in> carrier \<P>" shows "f (f x) .=\<^bsub>\<P>\<^esub> f x"
exec_d_eq_TypeError: fixes P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and t :: "'thread_id"   and \<sigma> :: "'addr option \<times> 'heap \<times> 'addr frame list" shows "(exec_d P t \<sigma> = TypeError) = (\<not> check P \<sigma>)"
adjacent_conv_insert: fixes x :: "'b set"   and y :: "'b set"   and v :: "'b" assumes "x \<sim> y"   and "v \<in> x - y" shows "x = insert v (x \<inter> y)"
strict_mono_on_eqD: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and x :: "'a"   and y :: "'a" assumes "strict_mono_on f A"   and "f x = f y"   and "x \<in> A"   and "y \<in> A" shows "y = x"
assms_1: shows "\<eta>1 \<approx>>\<bullet> \<eta>2"
dvd_unit_imp_unit: fixes a :: "'a"   and b :: "'a" assumes "a dvd b"   and "is_unit b" shows "is_unit a"
lift_gi_def: fixes G :: "('c, 'd) monoid_scheme"   and f :: "'e \<Rightarrow> 'c"   and gi :: "bool \<times> 'e" assumes "Group.group G" shows "group.lift_gi G f gi = (if fst gi then inv\<^bsub>G\<^esub> f (snd gi) else f (snd gi))"
inject: fixes x3 :: "'b"   and y3 :: "'b" shows "(dRETURN x3 = dRETURN y3) = (x3 = y3)"
evaluation_wrt_implicate_is_unsat: fixes S :: "'at Literal set set"   and C :: "'at Literal set" assumes "entails S C"   and "\<not> tautology C" shows "\<not> satisfiable (partial_evaluation S C)"
Lxy_not: fixes S :: "'b set"   and xs :: "'b list" assumes "S \<inter> set xs = {}" shows "Lxy xs S = []"
exhaust: fixes y :: "('a, 'b, 'c) trans" obtains x1 :: "'a" and x2 :: "'b" and x3 :: "'c" and x4 :: "'a" where "y = Trans x1 x2 x3 x4"
fi_pos: fixes i :: "'b" assumes "i \<in> L" shows "0 \<le> fi i"
Y: shows "Y \<in> sets (Pi\<^sub>M K M)"
neg_mask_twice: fixes x :: "'b word"   and n :: "nat"   and m :: "nat" shows "x AND NOT (mask n) AND NOT (mask m) = x AND NOT (mask (max n m))"
rep_eq: fixes x :: "'b poly"   and xa :: "'b poly" shows "coeff (x - xa) = (\<lambda>n. coeff x n - coeff xa n)"
ciins: shows "ci_app ci (ins ! pc) P h stk loc undefined undefined pc []"
distinct_5: fixes x4 :: "'b" shows "true\<^sub>n \<noteq> nprop\<^sub>n(x4)"
H: shows "(\<Union> (range (relations M')))\<^sup>* `` {t} \<subseteq> worlds M'"
ex: shows "\<epsilon> \<in> lists G\<^sub>+ \<and> concat \<epsilon> = \<epsilon>"
that: fixes p1 :: "('v \<times> 'w \<times> 'v) list"   and u :: "'v"   and w :: "'w"   and u' :: "'v"   and p2 :: "('v \<times> 'w \<times> 'v) list" assumes "p = p1 @ (u, w, u') # p2"   and "int_vertices p1 \<subseteq> W"   and "u \<in> W"   and "u' \<notin> W" shows "thesis"
Result: fixes b :: "'a"   and conv' :: "('b, 'a, 'c, 'd) converter"   and \<I>' :: "('c, 'd) \<I>"   and conv :: "('b, 'a, 'c, 'd) converter"   and a :: "'b"   and \<I> :: "('b, 'a) \<I>" assumes "(b, conv') \<in> results_gpv \<I>' (run_converter conv a)"   and "a \<in> outs_\<I> \<I>" shows "b \<in> results_converter \<I> \<I>' conv"
toplevel_summands_derivs_Star_nonempty: fixes xs :: "'b list"   and r :: "'b rexp" assumes "xs \<noteq> []" shows "toplevel_summands (derivs xs (Star r)) \<subseteq> {Times (derivs ys r) (Star r) |ys. \<exists>zs. ys \<noteq> [] \<and> zs @ ys = xs}"
comp_null_1: fixes f :: "'a" shows "B.null \<cdot> f = B.null"
assms_3: shows "ok (e\<lparr>a:X\<rparr>)"
NIA_def: fixes t1 :: "'b sparc_state"   and t2 :: "'b sparc_state" shows "NIA t1 t2 \<equiv> get_S (cpu_reg_val PSR t1) = 0 \<and> get_S (cpu_reg_val PSR t2) = 0 \<and> good_context t1 \<and> get_delayed_pool t1 = [] \<and> get_trap_set t1 = {} \<and> good_context t2 \<and> get_delayed_pool t2 = [] \<and> get_trap_set t2 = {} \<and> low_equal t1 t2"
\<P>_are_admissible: fixes p :: "(symbol \<times> character list) list"   and k :: "nat"   and u :: "nat" assumes "p \<in> \<P> k u" shows "admissible p"
sorted_list_of_set_UN_lessThan: fixes k :: "nat"   and A :: "nat \<Rightarrow> 'a set" assumes "strict_mono_sets {..<k} A"   and "\<And>i. i < k \<Longrightarrow> finite (A i)" shows "sorted_list_of_set (\<Union> (A ` {..<k})) = concat (map (sorted_list_of_set \<circ> A) (sorted_list_of_set {..<k}))"
SubstFormP_fresh_iff: fixes a :: "atom"   and v :: "tm"   and i :: "tm"   and x :: "tm"   and x' :: "tm" shows "a \<sharp> SubstFormP v i x x' = (a \<sharp> v \<and> a \<sharp> i \<and> a \<sharp> x \<and> a \<sharp> x')"
sec_pubk_less_def: fixes n :: "nat" shows "sec_pubk_less n \<equiv> insert (Spy, PubKey {Tok_PriK n, Rev_PriK}) s\<^sub>0"
advantage_def: fixes \<A> :: "(bool, 'message \<times> 'message + 'cipher, 'cipher option + 'message option) gpv" shows "advantage \<A> = \<bar>spmf (game \<A>) True - 1 / 2\<bar>"
xasset: shows "\<forall>x. x \<notin> stocks Mkt \<longrightarrow> (\<forall>Mkt2 p. coincides_on Mkt Mkt2 (stocks Mkt) \<and> prices Mkt2 x = pr \<and> portfolio p \<and> support_set p \<subseteq> stocks Mkt \<union> {x} \<longrightarrow> \<not> arbitrage_process Mkt2 p)"
lim_f: fixes n :: "nat" shows "(\<lambda>k. f (d k) (r n)) \<longlonglongrightarrow> lim (\<lambda>k. f (d k) (r n))"
set_restriction_commute: fixes f :: "'a set \<Rightarrow> 'a set"   and g :: "'a set \<Rightarrow> 'a set"   and I :: "'a set" assumes "set_restriction f"   and "set_restriction g" shows "f (g I) = g (f I)"
inj_def: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" shows "FactoredSystem.inj f A B \<equiv> (\<forall>x\<in>A. f x \<in> B) \<and> inj_on f A"
PiM_component: fixes i :: "'i" assumes "i \<in> I" shows "distr (Pi\<^sub>M I M) (M i) (\<lambda>\<omega>. \<omega> i) = M i"
rel_witness_spmf1: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'a spmf"   and q :: "'b spmf" assumes "rel_spmf A p q" shows "rel_spmf (\<lambda>a (a', b). a = a' \<and> A a' b) p (rel_witness_spmf A (p, q))"
that: assumes "\<forall>a. \<not> r a a" shows "thesis"
case_1: fixes f1 :: "'e"   and f2 :: "'f \<Rightarrow> 'e"   and f3 :: "'g \<Rightarrow> 'f \<Rightarrow> 'e" shows "(case Done of Done \<Rightarrow> f1 | Skip x \<Rightarrow> f2 x | Yield x xa \<Rightarrow> f3 x xa) = f1"
leI: fixes x :: "'a"   and y :: "'a" assumes "\<not> x < y" shows "y \<le> x"
red: shows "\<tau>reds1r P t h (es', xs) (es'', xs')"
Inf_eqI: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'b set"   and x :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq x i"   and "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> less_eq y i) \<Longrightarrow> less_eq y x" shows "Inf A = x"
simps_2: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and m :: "'b match_expr"   and ms :: "'b match_expr list"   and a :: "action"   and p :: "'c" shows "match_list \<gamma> (m # ms) a p = (if matches \<gamma> m a p then True else match_list \<gamma> ms a p)"
simp_exp_composed: fixes X :: "'a \<Rightarrow> 'c"   and Px :: "'c \<Rightarrow> real"   and f :: "'c \<Rightarrow> real" assumes "simple_distributed M X Px" shows "expectation (\<lambda>a. f (X a)) = (\<Sum>x\<in>X ` space M. f x * Px x)"
length_filter_le: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "length (filter P xs) \<le> length xs"
JN_component_iff: fixes I :: "'d set"   and F :: "'d \<Rightarrow> 'c program"   and H :: "'c program" shows "(JOIN I F \<le> H) = (\<forall>i\<in>I. F i \<le> H)"
tree_delete_in_traverse_pairs: fixes t :: "('c, 'd) tree"   and x :: "'c" assumes "tree_sorted t" shows "in_traverse_pairs (tree_delete x t) = remove_elt_pairs x (in_traverse_pairs t)"
finite_same_card_bij_on_ext_funcset: fixes A :: "'b set"   and B :: "'c set" assumes "finite A"   and "finite B"   and "card A = card B" shows "\<exists>f. f \<in> A \<rightarrow>\<^sub>E B \<and> bij_betw f A B"
max_list_Max: fixes xs :: "'t list" assumes "xs \<noteq> []" shows "ord_term_lin.max_list xs = ord_term_lin.Max (set xs)"
singleton_list_fusion_def: fixes gen :: "('b, 'c) generator"   and state :: "'c" shows "singleton_list_fusion gen state = (case list.unfoldr gen state of [] \<Rightarrow> False | [x] \<Rightarrow> True | x # aa # lista \<Rightarrow> False)"
simps_4: fixes v :: "int"   and p :: "int \<times> 'a" shows "del (Some v) (Branch2 Empty p Empty) = (case compare (Some v) p of EQUAL \<Rightarrow> Some (p, True, Empty) | _ \<Rightarrow> None)"
app_mono_def: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and app :: "nat \<Rightarrow> 'a \<Rightarrow> bool"   and n :: "nat"   and A :: "'a set" shows "app_mono r app n A = (\<forall>s p t. s \<in> A \<and> p < n \<and> s \<sqsubseteq>\<^bsub>r\<^esub> t \<longrightarrow> app p t \<longrightarrow> app p s)"
reachable_nodes_step'_1: fixes u :: "'a"   and g :: "'a ugraph"   and r :: "'a"   and v :: "'a" assumes "u \<in> reachable_nodes g r"   and "(u, v) \<in> edges g" shows "v \<in> reachable_nodes g r"
ltl_equivalence_axioms: shows "ltl_equivalence (\<sim>\<^sub>P)"
simps_1: fixes t :: "dbtm"   and u :: "dbtm" shows "quot_dbfm (DBMem t u) = Q_Mem (quot_dbtm t) (quot_dbtm u)"
vimage_image: fixes A :: "V" assumes "A \<subseteq>\<^sub>\<circ> \<D>\<^sub>\<circ> r" shows "elts (r `\<^sub>\<circ> A) = app r ` elts A"
restrict_vals_nonequiv1: fixes n :: "nat"   and vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant" assumes "vals_nonequiv K (Suc (Suc n)) vv" shows "vals_nonequiv K (Suc n) (compose {h. h \<le> Suc n} vv (skip 1))"
finite_range: fixes index :: "nat \<times> nat" shows "finite (local.range index)"
dist: shows "distinct (trd (xs @ ys) a # xs @ ys)"
nonsingular_def: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c" assumes "ell_field R" shows "ell_field.nonsingular R a b = (\<guillemotleft>4\<guillemotright>\<^bsub>R\<^esub> \<otimes>\<^bsub>R\<^esub> a [^]\<^bsub>R\<^esub> 3 \<oplus>\<^bsub>R\<^esub> \<guillemotleft>27\<guillemotright>\<^bsub>R\<^esub> \<otimes>\<^bsub>R\<^esub> b [^]\<^bsub>R\<^esub> 2 \<noteq> \<zero>\<^bsub>R\<^esub>)"
\<Delta>_simps_1: shows "seq (\<mu> \<star> tab\<^sub>0 r) r.tab"
is_partition_of_def: fixes P :: "'b set set"   and A :: "'b set" shows "P partitions A = (\<Union> P = A \<and> is_non_overlapping P)"
case_2: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a" shows "(case B of A \<Rightarrow> f1 | B \<Rightarrow> f2 | C \<Rightarrow> f3 | D \<Rightarrow> f4) = f2"
g: fixes l :: "nat list" shows "g l + sum_list l < ack kg (sum_list l)"
irreducible_not_unit: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and p :: "'c" assumes "class.comm_semiring_1 times one plus zero"   and "comm_semiring_1.irreducible times one zero p" shows "\<not> dvd.dvd times p one"
filter_mono: fixes F1 :: "'a filter"   and F2 :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "F1 \<le> F2"   and "f \<in> L F2 g" shows "f \<in> L F1 g"
image_add_ball: fixes b :: "'c"   and a :: "'c"   and r :: "real" shows "(+) b ` ball a r = ball (a + b) r"
irr: fixes U :: "'c set" assumes "RModule FG (\<cdot>) U \<and> U \<subseteq> V" shows "U = 0 \<or> U = V"
x': shows "arrow_from_functor (\<cdot>) (\<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>]) local.map a' d x'"
invariant_weakenE: fixes A :: "('a, 'b) automaton"   and PI :: "'b \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and QI :: "'b \<Rightarrow> bool" assumes "A \<TTurnstile> (PI \<rightarrow>) P"   and "\<And>s. P s \<Longrightarrow> Q s"   and "\<And>a. QI a \<Longrightarrow> PI a" shows "A \<TTurnstile> (QI \<rightarrow>) Q"
degreen_polyadd: fixes p :: "poly"   and n0 :: "nat"   and q :: "poly"   and n1 :: "nat"   and m :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1"   and "m \<le> max n0 n1" shows "degreen (p +\<^sub>p q) m \<le> max (degreen p m) (degreen q m)"
interior_subset_rel_interior: fixes S :: "'c set" shows "interior S \<subseteq> rel_interior S"
e_A_def: shows "e_A \<equiv> append_exec e_A' e"
pairwise_compat: fixes c1 :: "('a, 'b) chart"   and c2 :: "('a, 'b) chart" assumes "c1 \<in> charts"   and "c2 \<in> charts" shows "k-smooth_compat c1 c2"
Anno_code_i_i_i_o_def: shows "Anno_code_i_i_i_o \<equiv> \<lambda>x xa xb. pred.Pred (Anno_code x xa xb)"
1: shows "Partial_Recursive.list_encode (tuple_decode (pdec1 k) (pdec2 k)) = Suc (prod_encode (length xs, tuple_encode (length xs) (x # xs)))"
mk_circle_cmat_def: fixes a :: "complex"   and r :: "real" shows "mk_circle_cmat a r = (1, - a, - cnj a, a * cnj a - cor r * cor r)"
simps: fixes as :: "'a atom set"   and lb :: "nat \<Rightarrow> 'a option"   and ub :: "nat \<Rightarrow> 'a option" shows "as \<doteq> (lb, ub) = (\<forall>v. v \<Turnstile>\<^sub>a\<^sub>s as = v \<Turnstile>\<^sub>b (lb, ub))"
mat0_leastI: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "mat_gt_arc A (0\<^sub>m nr nc)"
rd_H1_H2_R_H1_H2: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" shows "R (\<not> (H1 \<circ> H2) (\<lambda>(A, A'). P (A, A'\<lparr>ok := False\<rparr>)) \<turnstile> (H1 \<circ> H2) (\<lambda>(A, A'). P (A, A'\<lparr>ok := True\<rparr>))) = (R \<circ> H1 \<circ> H2) P"
cut_greater_mem_iff: fixes x :: "'a"   and I :: "'a set"   and t :: "'a" shows "(x \<in> I \<down>> t) = (x \<in> I \<and> t < x)"
us: shows "us \<equiv> getFrN (map snd txs) (map fst txs) [scnj F] (length txs)"
M_Rcd: fixes fps :: "(char list \<times> pat) list"   and fs :: "(char list \<times> trm) list"   and ts :: "trm list" assumes "\<turnstile> fps [\<rhd>] fs \<Rightarrow> ts" shows "\<turnstile> PRcd fps \<rhd> Rcd fs \<Rightarrow> ts"
all_not_fin_conv: fixes A :: "'a fset" shows "(\<forall>x. x |\<notin>| A) = (A = {||})"
continuous_map_real_mult: fixes X :: "'a topology"   and f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "continuous_map X euclideanreal f"   and "continuous_map X euclideanreal g" shows "continuous_map X euclideanreal (\<lambda>x. f x * g x)"
tan_gt_zero: fixes x :: "real" assumes "0 < x"   and "x < pi / 2" shows "0 < tan x"
cong_gcd_eq: fixes a :: "'a"   and b :: "'a"   and m :: "'a" assumes "[a = b] (mod m)" shows "gcd a m = gcd b m"
bezout_coefficients_fst_snd: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Gcd :: "'a set \<Rightarrow> 'a"   and Lcm :: "'a set \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.euclidean_ring_gcd divide plus minus zero times one modulo euclidean_size unit_factor normalize gcd lcm Gcd Lcm uminus" shows "plus (times (fst (euclidean_ring_gcd.bezout_coefficients divide minus zero times one modulo unit_factor normalize a b)) a) (times (snd (euclidean_ring_gcd.bezout_coefficients divide minus zero times one modulo unit_factor normalize a b)) b) = gcd a b"
abs_apply_terms_def: fixes M :: "(('e, 'f, 'g) prot_fun, 'h) Term.term set"   and \<alpha> :: "nat \<times> bool \<Rightarrow> 'g set" shows "M \<cdot>\<^sub>\<alpha>\<^sub>s\<^sub>e\<^sub>t \<alpha> \<equiv> (\<lambda>t. t \<cdot>\<^sub>\<alpha> \<alpha>) ` M"
finite_field_factorization_main_uint32: fixes p :: "int"   and f :: "int poly"   and c :: "int"   and fs :: "int poly list" assumes "poly_mod_prime p"   and "p \<le> 65535"   and "finite_field_factorization_main p (finite_field_ops32 (uint32_of_int p)) f = (c, fs)"   and "poly_mod.square_free_m p f" shows "poly_mod.unique_factorization_m p f (c, mset fs) \<and> c \<in> {0..<p} \<and> (\<forall>fi\<in>set fs. set (coeffs fi) \<subseteq> {0..<p})"
splitTree_invpres_2: fixes s :: "('a, 'b) FingerTreeStruc"   and p :: "'b \<Rightarrow> bool"   and i :: "'b"   and l :: "('a, 'b) FingerTreeStruc"   and e :: "'a"   and a :: "'b"   and r :: "('a, 'b) FingerTreeStruc" assumes "FingerTreeStruc.ft_invar s"   and "\<not> p i"   and "p (i + FingerTreeStruc.annot s)"   and "FingerTreeStruc.splitTree p i s = (l, (e, a), r)" shows "FingerTreeStruc.ft_invar r"
enat3_cases: fixes y :: "enat"   and ya :: "enat"   and yb :: "enat" obtains nat :: "nat" and nata :: "nat" and natb :: "nat" where "y = enat nat__" and "ya = enat nata__" and "yb = enat natb"   | nat :: "nat" and nata :: "nat" where "y = enat natc__" and "ya = enat nataa__" and "yb = \<infinity>"   | nat :: "nat" and nata :: "nat" where "y = enat natd__" and "ya = \<infinity>" and "yb = enat natab__"   | nat :: "nat" where "y = enat nate__" and "ya = \<infinity>" and "yb = \<infinity>"   | nat :: "nat" and nata :: "nat" where "y = \<infinity>" and "ya = enat natf__" and "yb = enat nata"   | nat :: "nat" where "y = \<infinity>" and "ya = enat natg__" and "yb = \<infinity>"   | nat :: "nat" where "y = \<infinity>" and "ya = \<infinity>" and "yb = enat nat"   | "y = \<infinity>" and "ya = \<infinity>" and "yb = \<infinity>"
S'_def: shows "S' = {(t, u). (t, u) \<in> S \<and> (case (t, u) of (t, u) \<Rightarrow> t\<^sup>2 - int (u\<^sup>2 * D)) = k \<and> t mod \<bar>k\<bar> = l1 \<and> int u mod \<bar>k\<bar> = l2}"
inf: fixes l :: "nat"   and dl :: "nat \<Rightarrow> nat set"   and al :: "nat set"   and bl :: "nat \<times> nat \<Rightarrow> nat set"   and L :: "nat set" assumes "DF l = (dl, al, bl, L)" shows "infinite L"
simps_23: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "pdf_type \<Rightarrow> pdf_type \<Rightarrow> 'a" shows "(case BOOL of UNIT \<Rightarrow> f1 | BOOL \<Rightarrow> f2 | INTEG \<Rightarrow> f3 | REAL \<Rightarrow> f4 | PRODUCT x xa \<Rightarrow> f5 x xa) = f2"
unrest_pre_out\<alpha>: fixes b :: "('d, 'c) uexpr" shows "out\<alpha> \<sharp> \<lceil>b\<rceil>\<^sub><"
bind_scale_spmf: fixes r :: "real"   and p :: "'d spmf"   and f :: "'d \<Rightarrow> 'c spmf" assumes "r \<le> 1" shows "scale_spmf r p \<bind> f = p \<bind> (\<lambda>x. scale_spmf r (f x))"
ivl_2: fixes n :: "nat" shows "ivl (Suc n) = (i (fst (ivl n)) (snd (ivl n)) (f n), j__ (fst (ivl n)) (snd (ivl n)) (f n))"
cmr: shows "continuous_map (nsphere (n - Suc 0)) (nsphere (n - Suc 0)) f"
swap_Recv_Snapshot: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and d :: "('p, 's, 'm) configuration"   and ev' :: "('p, 's, 'm) event"   and e :: "('p, 's, 'm) configuration"   and d' :: "('p, 's, 'm) configuration"   and e' :: "('p, 's, 'm) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isRecv ev"   and "isSnapshot ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'"   and "occurs_on ev \<noteq> occurs_on ev'" shows "msgs e i = msgs e' i"
tSngl: assumes "invar l \<and> invar r \<and> lv = Suc (lvl r) \<and> lvl l = lvl r" shows "thesis"
neg_divideR_le_eq: fixes c :: "real"   and b :: "'a"   and a :: "'a" assumes "c < 0" shows "(b /\<^sub>R c \<le> a) = (c *\<^sub>R a \<le> b)"
bilinear_def: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "bilinear f = ((\<forall>x. linear (f x)) \<and> (\<forall>y. linear (\<lambda>x. f x y)))"
invar_2: shows "free_vars \<delta> \<subseteq> set (vs @ vs')"
dual_dual_ord: fixes x :: "'a"   and y :: "'a" shows "(\<partial> x \<le> \<partial> y) = (y \<le> x)"
wp3Conc: fixes a :: "((int \<times> int) set set, int) Combinators"   and p :: "((int \<times> int) set set, int) Combinators list" assumes "wellformed_policy3 (a # p)" shows "wellformed_policy3 p"
inf_right_isotone: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "z \<sqinter> x \<le> z \<sqinter> y"
abc: shows "a ^ 4 + b ^ 4 = ((z div gcd x y)\<^sup>2)\<^sup>2 \<and> a * b * (z div gcd x y)\<^sup>2 \<noteq> 0"
mkArr_in_hom: fixes A :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr set"   and B :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr set"   and F :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr \<Rightarrow> (('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr" assumes "A \<subseteq> S.Univ"   and "B \<subseteq> S.Univ"   and "F \<in> A \<rightarrow> B" shows "\<guillemotleft>S.mkArr A B F : S.mkIde A \<rightarrow>\<^sub>S S.mkIde B\<guillemotright>"
bigo_const3: fixes c :: "'a" assumes "c \<noteq> (0::'a)" shows "(\<lambda>x. 1::'a) \<in> O(\<lambda>x. c)"
cases: fixes a1 :: "nat \<Rightarrow> pdf_type"   and a2 :: "expr"   and a3 :: "pdf_type" assumes "a1 \<turnstile> a2 : a3" obtains \<Gamma> :: "nat \<Rightarrow> pdf_type" and x :: "nat" where "a1 = \<Gamma>__" and "a2 = Var x" and "a3 = \<Gamma>__ x"   | \<Gamma> :: "nat \<Rightarrow> pdf_type" and v :: "val" where "a1 = \<Gamma>'__" and "a2 = Val v" and "a3 = val_type v"   | \<Gamma> :: "nat \<Rightarrow> pdf_type" and e1 :: "expr" and t1 :: "pdf_type" and e2 :: "expr" and t2 :: "pdf_type" where "a1 = \<Gamma>''__" and "a2 = LET e1__ IN e2__" and "a3 = t2__" and "\<Gamma>''__ \<turnstile> e1__ : t1__" and "case_nat t1__ \<Gamma>''__ \<turnstile> e2__ : t2__"   | \<Gamma> :: "nat \<Rightarrow> pdf_type" and e :: "expr" and t :: "pdf_type" and oper :: "pdf_operator" and t' :: "pdf_type" where "a1 = \<Gamma>'''__" and "a2 = oper $$ e__" and "a3 = t'" and "\<Gamma>'''__ \<turnstile> e__ : t__" and "op_type oper t__ = Some t'"   | \<Gamma> :: "nat \<Rightarrow> pdf_type" and e1 :: "expr" and t1 :: "pdf_type" and e2 :: "expr" and t2 :: "pdf_type" where "a1 = \<Gamma>''''__" and "a2 = <e1a__ ,  e2a__>" and "a3 = PRODUCT t1 t2" and "\<Gamma>''''__ \<turnstile> e1a__ : t1" and "\<Gamma>''''__ \<turnstile> e2a__ : t2"   | \<Gamma> :: "nat \<Rightarrow> pdf_type" and e :: "expr" and dst :: "pdf_dist" where "a1 = \<Gamma>'''''__" and "a2 = Random dst e" and "a3 = dist_result_type dst" and "\<Gamma>'''''__ \<turnstile> e : dist_param_type dst"   | \<Gamma> :: "nat \<Rightarrow> pdf_type" and b :: "expr" and e1 :: "expr" and t :: "pdf_type" and e2 :: "expr" where "a1 = \<Gamma>''''''__" and "a2 = IF b THEN e1 ELSE e2" and "a3 = ta__" and "\<Gamma>''''''__ \<turnstile> b : BOOL" and "\<Gamma>''''''__ \<turnstile> e1 : ta__" and "\<Gamma>''''''__ \<turnstile> e2 : ta__"   | \<Gamma> :: "nat \<Rightarrow> pdf_type" and t :: "pdf_type" where "a1 = \<Gamma>" and "a2 = Fail t" and "a3 = t"
isTrue_prv_PPf_prf_or_neg: fixes "prf" :: "'proof"   and \<phi> :: "'fmla" assumes "prf \<in> proof"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "bprv (PPf (encPf prf) \<langle>\<phi>\<rangle>) \<or> bprv (neg (PPf (encPf prf) \<langle>\<phi>\<rangle>))"
lm104: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<forall>x\<in>X. f x = g x" shows "sum f X = sum g X"
diff_fun_spaceD: fixes f :: "'a \<Rightarrow> real" assumes "f \<in> sub.diff_fun_space" shows "diff_fun k (charts_submanifold S) f"
coprime_def': fixes p :: "'a"   and q :: "'a" shows "coprime p q \<equiv> \<forall>r. r dvd p \<longrightarrow> r dvd q \<longrightarrow> r dvd (1::'a)"
simps_6: shows "(\<infinity> \<le> -\<infinity>) = False"
bind_spmf_cong: fixes p :: "'r spmf"   and q :: "'r spmf"   and f :: "'r \<Rightarrow> 's spmf"   and g :: "'r \<Rightarrow> 's spmf" assumes "p = q"   and "\<And>x. x \<in> set_spmf q \<Longrightarrow> f x = g x" shows "p \<bind> f = q \<bind> g"