hcju/isabelleps huggingface dataset:
vsv_vimageI2: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> vreal_uminus" shows "-\<^sub>\<real> a \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vreal_uminus"
alw_smap: fixes P :: "'a stream \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and s :: "'b stream" shows "alw P (smap f s) = alw (\<lambda>x. P (smap f x)) s"
self_root: fixes u :: "'b list" shows "u \<in> u*"
closed: fixes g :: "nat \<Rightarrow> ordinal" assumes "\<forall>n. g n \<in> A" shows "oLimit g \<in> A"
disc_2: fixes xs :: "'f llist"   and b :: "'g" assumes "\<not> lnull xs" shows "\<not> is_TNil (tllist_of_llist b xs)"
degree_pcompose: fixes p :: "'a poly"   and q :: "'a poly" shows "degree (p \<circ>\<^sub>p q) = degree p * degree q"
prj_simps_1: shows "C.seq trg_trg.prj\<^sub>1 \<p>\<^sub>1[trg.leg0 \<cdot> trg_trg.prj\<^sub>0, f.leg1]"
simps_1: fixes f :: "'fsym"   and al :: "univ list" shows "TE_intF (Oldf f) al = intF f al"
induct: fixes x1 :: "('d, 'e) Term.term"   and x2 :: "('d, 'e) Term.term"   and P :: "('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> bool" assumes "x1 \<unrhd> x2"   and "\<And>t. P t t"   and "\<And>u ss t f. \<lbrakk>u \<in> set ss; u \<unrhd> t; P u t\<rbrakk> \<Longrightarrow> P (Fun f ss) t" shows "P x1 x2"
inject_2: fixes x2 :: "dB"   and y2 :: "dB" shows "(Pure x2 = Pure y2) = (x2 = y2)"
permutes_inj: fixes p :: "'e \<Rightarrow> 'e"   and S :: "'e set" assumes "p permutes S" shows "inj p"
rev_nth_snoc: fixes xs :: "'b list"   and x :: "'b"   and v :: "nat"   and y :: "'b" assumes "(xs @ [x]) !. Suc v = Some y" shows "xs !. v = Some y"
and_mask_arith: fixes w :: "'a word"   and n :: "nat" shows "w AND mask n = w * 2 ^ (size w - n) div 2 ^ (size w - n)"
index_mat_2: fixes i :: "nat"   and nr :: "nat"   and j :: "nat"   and nc :: "nat"   and g :: "nat \<Rightarrow> 'b Matrix.vec" assumes "i < nr"   and "j < nc" shows "mat\<^sub>r nr nc g $$ (i, j) = g i $v j"
aux: shows "X \<union> bisim = {(P, Q). (P, Q) \<in> X \<or> P \<sim> Q}"
InitDB_def: shows "InitDB = \<lparr>mbal = 0, bal = 0, inp = NotAnInput\<rparr>"
pr_simps_4: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "arr \<pp>\<^sub>1[a, b]"
num_reachable_def: fixes G :: "'b list_graph"   and v :: "'b" shows "FiniteListGraph.num_reachable G v = length (FiniteListGraph.succ_tran G v)"
xor_zero_left: fixes x :: "'a" shows "(0::'a) XOR x = x"
strictly_redundant_def: fixes C :: "'a Literal set"   and S :: "'a Literal set set" shows "strictly_redundant C S = (tautology C \<or> (\<exists>D. D \<in> S \<and> D \<subset> C))"
normalize: fixes cstart :: "'a list multiset"   and w :: "('b, 'c) label list"   and c' :: "'a list multiset"   and p :: "'b" assumes "(cstart, w, c') \<in> trcl (tr fg)"   and "cstart = {#[entry fg p]#}"   and "initialproc fg p" shows "\<exists>w'. ({#[entry fg p]#}, w', c') \<in> trcl (ntr fg)"
Imp_Neg_I: fixes H :: "fm set"   and A :: "fm"   and B :: "fm" assumes "H \<turnstile> A IMP B"   and "H \<turnstile> A IMP Neg B" shows "H \<turnstile> Neg A"
less_eq: fixes m :: "nat"   and n :: "nat" shows "((m, n) \<in> pred_nat\<^sup>+) = (m < n)"
items_auth_data_out: fixes x :: "rat"   and H :: "msg set"   and y :: "rat" assumes "Pri_AgrK x \<notin> items H"   and "Pri_AgrK y \<notin> items H" shows "items (insert (Auth_Data x y) H) = insert (Auth_Data x y) (items H)"
induct: fixes P :: "'a list \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "'a \<Rightarrow> bool"   and a2 :: "'a list" assumes "\<And>acc Pa. P acc Pa []"   and "\<And>acc Pa x xs. \<lbrakk>Pa x \<Longrightarrow> P (x # acc) Pa xs; \<not> Pa x \<Longrightarrow> P acc Pa xs\<rbrakk> \<Longrightarrow> P acc Pa (x # xs)" shows "P a0 a1 a2"
a0n_def: shows "a0n \<equiv> \<lparr>init = a0i_init, trans = a0n_trans, obs = id\<rparr>"
complement_closure: fixes x :: "i" assumes "\<exists>w. \<not> O w x" shows "\<exists>z. \<forall>w. P w z = (\<not> O w x)"
iso_normal_subgroup: fixes G :: "('c, 'd) monoid_scheme"   and F :: "('e, 'f) monoid_scheme"   and H :: "'c set"   and \<phi> :: "'c \<Rightarrow> 'e" assumes "Group.group G"   and "Group.group F"   and "H \<lhd> G"   and "\<phi> \<in> Group.iso G F" shows "\<phi> ` H \<lhd> F"
list_all2_lengthD: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and xs :: "'g list"   and ys :: "'h list" assumes "list_all2 P xs ys" shows "length xs = length ys"
mulex_on_all_strict: fixes X :: "'b multiset"   and A :: "'b set"   and Y :: "'b multiset"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "X \<noteq> {#}"   and "X \<in> multisets A"   and "Y \<in> multisets A"   and "\<forall>y. y \<in># Y \<longrightarrow> (\<exists>x. x \<in># X \<and> P y x)" shows "mulex_on P A Y X"
classes_above_subcls2: fixes C :: "char list"   and P :: "'b prog"   and P' :: "'b prog"   and C' :: "char list" assumes "classes_above P C \<inter> classes_changed P P' = {}"   and "P' \<turnstile> C \<preceq>\<^sup>* C'" shows "P \<turnstile> C \<preceq>\<^sup>* C'"
inverse_divide: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.field divide inverse times one plus zero minus uminus" shows "inverse (divide a b) = divide b a"
plambda_eta: fixes f :: "('a, 'b) pfun" shows "plambda (\<lambda>x. x \<in> pdom f) (pfun_app f) = f"
disc_84: fixes x7 :: "ipt_l4_ports" shows "\<not> is_MultiportPorts (Dst_Ports x7)"
matrix_vector_right_distrib: fixes A :: "(('b, 'd) vec, 'c) vec"   and x :: "('b, 'd) vec"   and y :: "('b, 'd) vec" shows "A *v (x + y) = A *v x + A *v y"
zero_ext_f: fixes y :: "'a" assumes "y \<in> FG - supp f" shows "f y * h (- y + x) = (0::'b)"
coeff_pCons: fixes x :: "'a"   and xa :: "'a poly" shows "coeff (pCons x xa) = case_nat x (coeff xa)"
ext_inject: fixes g_V :: "'a set"   and g_E :: "('a \<times> 'a) set"   and g_V0 :: "'a set"   and more :: "'b"   and g_V' :: "'a set"   and g_E' :: "('a \<times> 'a) set"   and g_V0' :: "'a set"   and more' :: "'b" shows "(\<lparr>g_V = g_V, g_E = g_E, g_V0 = g_V0, \<dots> = more\<rparr> = \<lparr>g_V = g_V', g_E = g_E', g_V0 = g_V0', \<dots> = more'\<rparr>) = (g_V = g_V' \<and> g_E = g_E' \<and> g_V0 = g_V0' \<and> more = more')"
openin_Int_open: fixes U :: "'c set"   and S :: "'c set"   and T :: "'c set" assumes "openin (top_of_set U) S"   and "open T" shows "openin (top_of_set U) (S \<inter> T)"
m_mod: shows "(n * 10 + char_to_digit c) mod 10 = char_to_digit c"
IN_g_A: fixes x :: "'v" assumes "x \<in> A \<Gamma>" shows "d_IN (\<lambda>(x, y). g (\<langle>y\<rangle>, \<langle>x\<rangle>)) x = d_OUT g \<langle>x\<rangle> - g (\<langle>x\<rangle>, vertex.SINK)"
Abs_mem_inverse: fixes y :: "uint8 list" assumes "y \<in> UNIV" shows "Rep_mem (Abs_mem y) = y"
mahler_measure_monic_ge_0: fixes p :: "complex poly" shows "0 \<le> mahler_measure_monic p"
minus_lt: fixes a :: "'a"   and b :: "'a" shows "(a < b) = (a - b < (0::'a))"
induct: fixes P :: "('c, 'd) rose_tree\<^sub>m \<Rightarrow> ('c, 'd) rose_tree\<^sub>m \<Rightarrow> bool"   and a0 :: "('c, 'd) rose_tree\<^sub>m"   and a1 :: "('c, 'd) rose_tree\<^sub>m" assumes "\<And>x y. (\<And>a b. \<lbrakk>a \<in> set1_blindable\<^sub>m x \<bind> snds \<bind> set; b \<in> set1_blindable\<^sub>m y \<bind> snds \<bind> set\<rbrakk> \<Longrightarrow> P a b) \<Longrightarrow> P (Tree\<^sub>m x) (Tree\<^sub>m y)" shows "P a0 a1"
row_transpose: fixes i :: "'d"   and A :: "(('b, 'd) vec, 'c) vec" shows "row i (Finite_Cartesian_Product.transpose A) = column i A"
less_TC_imp_not_le: fixes x :: "V"   and a :: "V" assumes "x \<sqsubset> a" shows "\<not> a \<subseteq>\<^sub>\<circ> x"
analz_Un_implConfidSet: fixes G :: "msg set"   and H :: "msg set"   and Ag :: "(agent \<times> agent) set" assumes "Enc_keys_clean (G \<union> H)" shows "analz ({asym.implConfid A B M |A B M. (A, B) \<in> Ag \<and> M \<in> G} \<union> H) \<subseteq> synth (analz (G \<union> H)) \<union> - payload"
th: shows "(0 < real_of_int a + b) = (real_of_int (- a) + - b < 0)"
convex_imp_simply_connected: fixes S :: "'a set" assumes "convex S" shows "simply_connected S"
less_imp_le: fixes x :: "'b list"   and y :: "'b list" assumes "strict_prefix x y" shows "prefix x y"
split_at_it_def: fixes n :: "nat"   and xs :: "'b list" shows "split_at_it n xs = split_at_it' [] n xs"
dgrad_sig_set_closed_monom_mult: fixes d :: "'a \<Rightarrow> nat"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and j :: "nat"   and c :: "'b" assumes "dickson_grading d"   and "d t \<le> dgrad_max d"   and "p \<in> dgrad_sig_set' j d" shows "monom_mult c t p \<in> dgrad_sig_set' j d"
state_sim_def: fixes s :: "'b"   and A :: "('b \<Rightarrow> bool) set"   and t :: "'b" shows "(s ~A~ t) = (\<forall>p\<in>A. p s = p t)"
tick_count_f_suc_suc: fixes f :: "nat \<Rightarrow> nat"   and sub :: "'b run"   and r :: "'b run"   and c :: "clock"   and n :: "nat" assumes "dilating f sub r" shows "tick_count r c (f (Suc n)) = (if hamlet (Rep_run r (f (Suc n)) c) then Suc (tick_count r c (f n)) else tick_count r c (f n))"
map_ext: fixes xs :: "'g list"   and f :: "'g \<Rightarrow> 'h"   and g :: "'g \<Rightarrow> 'h" assumes "\<And>x. x \<in> set xs \<longrightarrow> f x = g x" shows "map f xs = map g xs"
inj_imp_image_ring_is_ring: fixes R :: "('a, 'b) ring_scheme"   and f :: "'a \<Rightarrow> 'c" assumes "ring R"   and "inj_on f (carrier R)" shows "ring (image_ring f R)"
simps_1: fixes s :: "'d stream" shows "s !! 0 = shd s"
"1_6": fixes j :: "nat" assumes "i \<le> j"   and "j < jj"   and "prime j" shows "j \<in> set is"
one_fract_def: shows "1 \<equiv> abs_fract (1::'a, 1::'a)"
min: fixes v :: "'t" assumes "v \<prec>\<^sub>t u" shows "v \<notin> lt ` M"
lem_e2: fixes x :: "real" assumes "x \<in> {x. poly p x = 0}" shows "card {s \<in> consistent_sign_vec_copr qs ` {x. poly p x = 0} \<inter> {s. z I s = - 1}. consistent_sign_vec_copr qs x = s} = (if poly (prod_list (retrieve_polys qs I)) x < 0 then 1 else 0)"
continuous_closedin_preimage_eq: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "continuous_on S f = (\<forall>T. closed T \<longrightarrow> closedin (top_of_set S) (S \<inter> f -` T))"
that: shows "\<forall>x\<in>set_spmf q. trace_eq_simcl S p (f x)"
prems_1: shows "t \<in> trms\<^sub>s\<^sub>s\<^sub>t (s # S)"
hyp2_axiom10_false: shows "\<not> (\<forall>a b c d t. B\<^sub>K a d t \<and> B\<^sub>K b d c \<and> a \<noteq> d \<longrightarrow> (\<exists>x y. B\<^sub>K a b x \<and> B\<^sub>K a c y \<and> B\<^sub>K x t y))"
mult_eq_0_iff: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and times :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and a :: "'g"   and b :: "'g" assumes "class.semiring_no_zero_divisors plus zero times" shows "(times a b = zero) = (a = zero \<or> b = zero)"
existential_lookup_sem: fixes fml :: "nat fml"   and qs :: "rat poly list"   and signs :: "rat list list" assumes "\<And>i. i \<in> set_fml fml \<Longrightarrow> i < length qs"   and "set signs = consistent_sign_vectors qs UNIV" shows "(\<exists>x. lookup_sem fml (map (\<lambda>p. rpoly p x) qs)) = (find (lookup_sem fml) signs \<noteq> None)"
normal_moment_odd: fixes \<sigma> :: "real"   and \<mu> :: "real"   and k :: "nat" assumes "0 < \<sigma>" shows "has_bochner_integral lborel (\<lambda>x. normal_density \<mu> \<sigma> x * (x - \<mu>) ^ (2 * k + 1)) 0"
reindex_bij_betw: fixes h :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and T :: "'d set"   and g :: "'d \<Rightarrow> 'e" assumes "bij_betw h S T" shows "(\<Prod>x\<in>S. g (h x)) = prod g T"
simps: fixes xs :: "('dyn, 'd, 'e, 'f, 'opinl, 'opubx, 'num, 'bool) instr list"   and ar :: "nat" shows "generalize_fundef (Fundef xs ar) = Fundef (map generalize_instr xs) ar"
alt_nondet_def: fixes return :: "'c \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and union :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and m1 :: "('a, 'b) nondetT"   and m2 :: "('a, 'b) nondetT" shows "nondetM_base.alt_nondet return bind union m1 m2 = NondetT (bind (run_nondet m1) (\<lambda>A. bind (run_nondet m2) (\<lambda>B. return (union A B))))"
final: fixes r :: "'a" assumes "I r s"   and "\<not> C r s" shows "Q (Some r)"
fcomp_def: fixes f :: "'b \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c" shows "f ;; g = (\<lambda>x. g (f x))"
QReachableTrans: fixes trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and cfg2 :: "('a, 'c, 'b) configuration"   and Q :: "'a set"   and cfg3 :: "('a, 'c, 'b) configuration"   and cfg1 :: "('a, 'c, 'b) configuration" assumes "asynchronousSystem.qReachable trans sends cfg2 Q cfg3"   and "asynchronousSystem.qReachable trans sends cfg1 Q cfg2" shows "asynchronousSystem.qReachable trans sends cfg1 Q cfg3"
Hge0: fixes Z :: "'a set"   and x :: "'a" assumes "Z \<in> \<C>"   and "x \<in> U - S" shows "0 \<le> H Z x"
prod_eqI: fixes p :: "'e \<times> 'f"   and q :: "'e \<times> 'f" assumes "fst p = fst q"   and "snd p = snd q" shows "p = q"
is_none_code_2: fixes x :: "'m" shows "Option.is_none (Some x) = False"
root_2: shows "rsquarefree ([:- a, 1::'a:] * p * q)"
\<pi>_fix: fixes T :: "(symbol \<times> character list) set"   and k :: "nat"   and I :: "item set" shows "Scan T k (Complete k (Predict k (\<pi> k T I))) = \<pi> k T I"
ping: shows "\<forall>i\<in>I. act (g i) ` (X - Xout i) \<subseteq> Xin i"
assms: fixes \<Psi>' :: "'b"   and Q :: "('a, 'b, 'c) psi" shows "(\<Psi>', Q, Q) \<in> Rel"
Hermite_of_row_i_works: fixes a :: "'a"   and i :: "'a"   and A :: "(('b, 'c) vec, 'a) vec"   and ass :: "'b \<Rightarrow> 'b"   and res :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'c" assumes "a < i"   and "\<not> is_zero_row i A" shows "Hermite_of_row_i ass res A i $ a $ b = row_add (mult_row A i (ass (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b))) div A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b)))) a i ((res (mult_row A i (ass (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b))) div A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b))) $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b))) (mult_row A i (ass (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b))) div A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b))) $ a $ (LEAST n. A $ i $ n \<noteq> (0::'b))) - mult_row A i (ass (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b))) div A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b))) $ a $ (LEAST n. A $ i $ n \<noteq> (0::'b))) div mult_row A i (ass (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b))) div A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b))) $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b))) $ a $ b"
hyp_override: fixes dynM :: "qtname \<times> methd" assumes "dynmethd G statC dynC sig = Some dynM"   and "dynM \<noteq> statM"   and "G \<turnstile> qmdecl sig dynM overrides qmdecl sig statM" shows "thesis"
correctCompKS_Secrets: fixes C :: "specID" assumes "correctCompositionKS C" shows "correctCompositionSecrets C"
sectionI: fixes g :: "'b"   and f :: "'b" assumes "B.ide (g \<cdot>\<^sub>B f)" shows "B.section f"
const_on_def: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and x :: "'b" shows "const_on f S x = (\<forall>y\<in>S. f y = x)"
gsurjec_def: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g" shows "gsurj\<^bsub>F,G\<^esub> f = (f \<in> gHom F G \<and> surj_to f (carrier F) (carrier G))"
list_all2I: fixes a :: "'a list"   and b :: "'b list"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>x\<in>set (zip a b). case x of (x, xa) \<Rightarrow> P x xa"   and "length a = length b" shows "list_all2 P a b"
finfun_rec_const: fixes c :: "'c" shows "finfun_rec (finfun_const \<circ> finfun_const) (\<lambda>(a, b) c f. finfun_update f a (finfun_update (finfun_apply f a) b c)) (finfun_const c) = (finfun_const \<circ> finfun_const) c"
refl: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "reflexive A less_eq"   and "x \<in> A" shows "less_eq x x"
word_unat_Rep_inject1: fixes x :: "'a word" shows "(unat x = unat 1) = (x = 1)"
assert_false: shows "{\<cdot> \<bottom> } = \<bottom>"
lm067: fixes R :: "('d \<times> 'c) set"   and X :: "'d set" shows "(R || X) `` X = R `` X"
assms: fixes i :: "nat" assumes "i \<le> n" shows "ideal (I i) R"
concat_map_singleton: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "concat (map (\<lambda>x. [f x]) xs) = map f xs"
sigma_finite_measure_axioms: fixes s :: "'a" shows "sigma_finite_measure (K1.T s)"
numgcd_def: fixes t :: "Ferrack.num" shows "numgcd t = numgcdh t (maxcoeff t)"
wlsAbs_Abs_varSort: fixes X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and s :: "'sort"   and xs' :: "'varSort"   and x :: "'var" assumes "good X"   and "wlsAbs (xs, s) (Abs xs' x X)" shows "xs = xs'"
intros_2: fixes \<phi>' :: "'val"   and g :: "'g"   and \<phi> :: "'val"   and \<phi>'' :: "'val" assumes "\<phi>' \<in> reachable g \<phi>"   and "phiArg g \<phi>' \<phi>''"   and "unnecessaryPhi g \<phi>''" shows "\<phi>'' \<in> reachable g \<phi>"
sorted_auxlist0: shows "sorted_wrt (\<lambda>x y. fst y < fst x) auxlist0"
infinite_finite_induct: fixes P :: "'h set \<Rightarrow> bool"   and A :: "'h set" assumes "\<And>A. infinite A \<Longrightarrow> P A"   and "P {}"   and "\<And>x F. \<lbrakk>finite F; x \<notin> F; P F\<rbrakk> \<Longrightarrow> P (insert x F)" shows "P A"
skip_im_Tr4: fixes x :: "nat"   and n :: "nat" assumes "x \<le> Suc n"   and "0 < x" shows "x - Suc 0 \<le> n"
zmset_add: fixes A :: "'b zmultiset"   and a :: "'b" obtains B :: "'b zmultiset" where "A = add_zmset a B"
assms_4: fixes x :: "'node" assumes "x \<in> set (tl ns)" shows "v \<notin> allDefs g x"
BIGTHETA_FUN_tag: fixes f :: "'a \<Rightarrow> 'b" shows "\<Theta>(f) = BIGTHETA_FUN f"
iso_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ide a"   and "ide b"   and "ide c" shows "local.iso (MC.assoc a b c)"
relcompI: fixes r :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'b"   and b :: "'d"   and s :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and c :: "'e" assumes "r a b"   and "s b c" shows "(r OO s) a c"
convex_on_continuous: fixes s :: "'b set"   and f :: "'b \<Rightarrow> real" assumes "open s"   and "convex_on s f" shows "continuous_on s f"
If_le: fixes b :: "bexp"   and c1 :: "'a acom"   and c2 :: "'a acom"   and S :: "'a"   and c :: "'a acom" shows "(IF b THEN c1 ELSE c2 {S} \<le> c) = (\<exists>c1' c2' S'. c = IF b THEN c1' ELSE c2' {S'} \<and> c1 \<le> c1' \<and> c2 \<le> c2' \<and> S \<le> S')"
frange_small: fixes r :: "V" shows "small {b. \<exists>a. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r}"
C_eq_compileQ: fixes p :: "((int \<times> int) set set, int) Combinators"   and l :: "(int \<times> int) set set set list" assumes "DenyAll \<in> set (policy2list p)"   and "all_in_list (policy2list p) l"   and "allNetsDistinct (policy2list p)" shows "C (list2FWpolicy (removeAllDuplicates (insertDenies (separate (qsort (removeShadowRules2 (remdups (rm_MT_rules C (insertDeny (removeShadowRules1 (policy2list p)))))) l))))) = C p"
map_comp: fixes g :: "'e \<Rightarrow> 'd"   and f :: "'f \<Rightarrow> 'e"   and v :: "'f rexp" shows "map_rexp g (map_rexp f v) = map_rexp (g \<circ> f) v"
diff_list_set_remove_Nil: fixes xs :: "'a list" shows "diff_list xs [] = xs"
below_in_refl: fixes x :: "'a"   and P :: "'a set" assumes "x \<in> P" shows "x \<in> P .\<^bold>\<le> x"
assms_2: shows "\<Psi> \<simeq> \<Psi>'"
lens_override_put_right_in: fixes A :: "'d \<Longrightarrow> 'e"   and X :: "'f \<Longrightarrow> 'e"   and S\<^sub>1 :: "'e"   and S\<^sub>2 :: "'e"   and v :: "'f" assumes "vwb_lens A"   and "X \<subseteq>\<^sub>L A" shows "S\<^sub>1 \<triangleleft>\<^bsub>A\<^esub> put\<^bsub>X\<^esub> S\<^sub>2 v = put\<^bsub>X\<^esub> (S\<^sub>1 \<triangleleft>\<^bsub>A\<^esub> S\<^sub>2) v"
closure_Int_open_eq_empty: fixes S :: "'c set"   and T :: "'c set" assumes "open S" shows "(closure T \<inter> S = {}) = (T \<inter> S = {})"
x: shows "x \<in> map \<pi> ` (lang (Suc n) (ENC (Suc n) (FOV \<phi>)) - {[]})"
prems_1: shows "ccw'.sortedP 0 (P # Ps)"
tab_in_hom_2: shows "\<guillemotleft>\<rho> : g \<Rightarrow> (g \<star> f\<^sup>*) \<star> f\<guillemotright>"
numer_cf4_pos: fixes x :: "real" shows "0 < numer_cf4 x"
simple_quadratic_hensel_step_def: fixes q :: "int"   and S :: "int poly"   and T :: "int poly"   and D :: "int poly"   and H :: "int poly" shows "local.simple_quadratic_hensel_step q S T D H = (let U = poly_mod.Mp q (sdiv_poly (C - D * H) q); (A, B) = dupe_monic_dynamic q D H S T U; D' = D + Polynomial.smult q B in Let (H + Polynomial.smult q A) (Pair D'))"
ln_Gamma_bounds_aux_2: fixes x :: "real" assumes "1 \<le> x" shows "ln_Gamma x \<le> c + (x - 1 / 2) * ln x - x + 1 / (12 * x)"
ring_hom_one: fixes h :: "'d \<Rightarrow> 'e"   and R :: "('d, 'f) ring_scheme"   and S :: "('e, 'g) ring_scheme" assumes "h \<in> ring_hom R S" shows "h \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>S\<^esub>"
effect_bindE: fixes f :: "'c Heap"   and g :: "'c \<Rightarrow> 'b Heap"   and h :: "heap"   and h'' :: "heap"   and r' :: "'b" assumes "effect (f \<bind> g) h h'' r'" obtains h' :: "heap" and r :: "'c" where "effect f h h' r" and "effect (g r) h' h'' r'"
fmpt_factor: fixes proj :: "'a \<Rightarrow> 'c"   and M2 :: "'c measure"   and T2 :: "'c \<Rightarrow> 'c" assumes "mpt_factor proj M2 T2" shows "fmpt M2 T2"
good: shows "qGood (pick \<circ> asTerm) (qOp delta (pickInp inp) (pickBinp binp)) #[[pickE rho]]"
subterm_wpo_ns_arg: fixes i :: "nat"   and f :: "'f"   and ss :: "('f, 'v) Term.term list" assumes "i \<in> set (\<sigma> (f, length ss))" shows "Fun f ss \<succeq> ss ! i"
iMODb_Suc_insert_conv: fixes r :: "nat"   and m :: "nat"   and c :: "nat" shows "insert (r + m * Suc c) [ r, mod m, c ] = [ r, mod m, Suc c ]"
simps: fixes x :: "name"   and r :: "tm"   and d :: "tm"   and y :: "name"   and z :: "name" assumes "atom x \<sharp> (r, d, y, z)"   and "atom y \<sharp> (r, d, z)"   and "atom z \<sharp> (r, d)" shows "HDomain_Incl r d = All2 x d (SyntaxN.Ex y (SyntaxN.Ex z (Var z IN r AND Var z EQ HPair (Var x) (Var y))))"
ext_eq_len_eq: fixes v :: "view"   and v' :: "view"   and ts :: "traffic"   and c :: "cars" shows "ext v = ext v' \<and> own v = own v' \<longrightarrow> len v ts c = len v' ts c"
remove_vertex_path_from_to_add: fixes G :: "('c, 'd) Graph_scheme"   and x :: "'c"   and v :: "'c"   and xs :: "'c list"   and w :: "'c" assumes "Digraph G"   and "v \<leadsto>xs\<leadsto>\<^bsub>Digraph.remove_vertex G x\<^esub> w" shows "v \<leadsto>xs\<leadsto>\<^bsub>G\<^esub> w"
id_dom_cod_1: fixes CC :: "('d, 'e, 'f) category_scheme"   and A :: "'d" assumes "category CC"   and "A \<in> Ob\<^bsub>CC\<^esub>" shows "Dom\<^bsub>CC\<^esub> (Id\<^bsub>CC\<^esub> A) = A"
compose_extensional: fixes A :: "'a set"   and f :: "'c \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c" shows "f \<circ> g \<down> A \<in> extensional A"
ccInf_less_iff: fixes S :: "'b set"   and a :: "'b" assumes "countable S" shows "(Inf S < a) = (\<exists>x\<in>S. x < a)"
vsv_valneq_v11I: assumes "\<And>x y. \<lbrakk>x \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r; y \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r; x \<noteq> y\<rbrakk> \<Longrightarrow> r\<lparr>x\<rparr> \<noteq> r\<lparr>y\<rparr>" shows "v11 r"
power_poly_sum_mod_ring: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'a mod_ring poly" assumes "finite A" shows "sum f A ^ CARD('a) = (\<Sum>i\<in>A. f i ^ CARD('a))"
assms_7: shows "poincare_distance y u = poincare_distance y v"
interp_msg_def: fixes msg :: "'a \<times> ('a, 'b) operation"   and state :: "'b \<Rightarrow> 'a set \<times> 'a set" shows "interp_msg msg state \<equiv> \<langle>snd msg\<rangle> state"
D'_def: shows "D' = ChamberComplex.closest_supchamber A F C"
N_of_norm_space: shows "space\<^sub>N N_of_norm = UNIV"
SecurityInvariant_preliminaries_axioms: shows "SecurityInvariant_preliminaries sinvar"
in_pair_Int_pairs: fixes p :: "'e \<times> 'e"   and A :: "('e \<times> 'e) set"   and B :: "('e \<times> 'e) set" shows "(p \<in>\<^sub>p A \<inter>\<^sub>p B) = (p \<in>\<^sub>p A \<and> p \<in>\<^sub>p B)"