hcju/isabelleps huggingface dataset:
OclForall_cong': fixes X :: "'e state \<times> 'e state \<Rightarrow> Set('d)"   and \<tau> :: "'e state \<times> 'e state"   and P :: "('e state \<times> 'e state \<Rightarrow> 'd) \<Rightarrow> 'e state \<times> 'e state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and Q :: "('e state \<times> 'e state \<Rightarrow> 'd) \<Rightarrow> 'e state \<times> 'e state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and R :: "('e state \<times> 'e state \<Rightarrow> 'd) \<Rightarrow> 'e state \<times> 'e state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<And>x. \<lbrakk>x \<in> \<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (X \<tau>)\<rceil>\<rceil>; \<tau> \<Turnstile> P (\<lambda>\<tau>. x); \<tau> \<Turnstile> Q (\<lambda>\<tau>. x)\<rbrakk> \<Longrightarrow> \<tau> \<Turnstile> R (\<lambda>\<tau>. x)"   and "\<tau> \<Turnstile> UML_Set.OclForall X P"   and "\<tau> \<Turnstile> UML_Set.OclForall X Q" shows "\<tau> \<Turnstile> UML_Set.OclForall X R"
ltln_strong_to_weak_2: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" assumes "w \<Turnstile>\<^sub>n \<phi> M\<^sub>n \<psi>" shows "w \<Turnstile>\<^sub>n \<phi> R\<^sub>n \<psi>"
obs: shows "0 \<in> {- r / 2<--<r / 2}"
modular_2_var: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x ; y \<cdot> z \<le> (x \<cdot> z ; y\<^sup>\<smile>) ; y"
conga_right_comm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A B C CongA D E F" shows "A B C CongA F E D"
sum_list_neutral: fixes xs :: "'a list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> x = (0::'a)" shows "sum_list xs = (0::'a)"
hyps_4: shows "cs\<^bsup>path \<sigma>\<^esup> n = cs\<^bsup>path \<sigma>'\<^esup> n'"
valid_hd_Rcons: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'a"   and tr :: "('a, 'b, 'c) trans list"   and tran :: "('a, 'b, 'c) trans" assumes "IO_Automaton.valid step tr" shows "hd (tr ## tran) = hd tr"
no_loops': fixes u :: "'a"   and v :: "'a" assumes "(u, v) \<in> parcs slim" shows "u \<noteq> v"
subdg_is_arr_vsubset: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>\<BB>\<^esub> b" shows "f : a \<mapsto>\<^bsub>\<CC>\<^esub> b"
of_circline_cmat_def: fixes H :: "complex \<times> complex \<times> complex \<times> complex" shows "of_circline_cmat H = (if pos_oriented_cmat H then H else opposite_ocircline_cmat H)"
zip_map2: fixes xs :: "'b list"   and f :: "'d \<Rightarrow> 'c"   and ys :: "'d list" shows "zip xs (map f ys) = map2 (\<lambda>x y. (x, f y)) xs ys"
uedge_eq_iff: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" shows "(uedge (a, b) = uedge (c, d)) = (a = c \<and> b = d \<or> a = d \<and> b = c)"
R2: shows "Y Midpoint P P2 \<and> Col A B Y"
boundedI: fixes a :: "'a word"   and b :: "'a word"   and c :: "'a word" assumes "a \<le>s b"   and "a \<le>s c" shows "a \<le>s signed.min b c"
cases: fixes x :: "('a \<Rightarrow> 'b) \<times> 'a \<times> 'a" obtains f :: "'a \<Rightarrow> 'b" and xa :: "'a" and y :: "'a" where "x = (f, xa, y)"
leq_trans: fixes x :: "'a Heap"   and y :: "'a Heap"   and z :: "'a Heap" assumes "Heap_ord x y"   and "Heap_ord y z" shows "Heap_ord x z"
b'_insertion': shows "insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse (isolate_variable_sparse p var (Suc 0)) var 0) = b'"
min_zero: fixes m :: "nat"   and x :: "nat"   and k :: "nat"   and T :: "nat \<Rightarrow> nat" assumes "LoadBalancing m"   and "x \<in> {1..k}"   and "T x = 0" shows "T (min\<^sub>k T k) = 0"
B: shows "\<forall>x\<in>D -` {u}. (ipurge_tr_rev I D u xs @ [x] \<in> T) = (xs @ [x] \<in> T)"
lnorm_triangle_fun: fixes f :: "'a \<Rightarrow> real"   and p :: "real"   and M :: "'a measure"   and g :: "'a \<Rightarrow> real" assumes "f \<in> lspace M (ennreal p)"   and "g \<in> lspace M (ennreal p)"   and "1 \<le> p" shows "lnorm M p (f + g) \<le> lnorm M p f + lnorm M p g"
ivl_disj_un_12: fixes l :: "'a"   and u :: "'a" assumes "l < u" shows "{l<..<u} \<union> {u..} = {l<..}"
at_set_avoiding: fixes Xs :: "'a set"   and c :: "'b" assumes "at TYPE('a)"   and "finite Xs"   and "finite (supp c)" obtains pi :: "('a \<times> 'a) list" where "(pi \<bullet> Xs) \<sharp>* c" and "set pi \<subseteq> Xs \<times> pi \<bullet> Xs"
L'_def: fixes F' :: "'b filter"   and g' :: "'b \<Rightarrow> 'c" shows "O[F'](g') = {f. \<exists>c>0. \<forall>\<^sub>F x in F'. norm (f x) \<le> c * norm (g' x)}"
fps_of_ratfps_mult: fixes x :: "'b ratfps"   and y :: "'b ratfps" shows "fps_of_ratfps (x * y) = fps_of_ratfps x * fps_of_ratfps y"
DW_def: shows "DW = (\<lambda>n. return_time_function A -` {n} \<inter> induced_map A --` W)"
dvd_gcd_0_iff_1: fixes x :: "'a"   and a :: "'a" shows "(x dvd gcd (0::'a) a) = (x dvd a)"
as_good_as_sameIff: fixes z :: "'a"   and y :: "'a" shows "(z \<in> as_good_as y carrier relation) = (z \<succeq> y \<and> y \<succeq> z)"
rel_gpv''_map_gpv'2: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and C :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and h :: "'f \<Rightarrow> 'e"   and gpv :: "('a, 'c, 'e) gpv"   and gpv' :: "('b, 'd, 'e) gpv" assumes "rel_gpv'' A C (eq_on (range h)) gpv gpv'" shows "rel_gpv'' A C (BNF_Def.Grp UNIV h)\<inverse>\<inverse> gpv (map_gpv' id id h gpv')"
C_is: shows "C = {(State k (index (\<Pi>\<^sub>\<V>) v))\<inverse>, (State (Suc k) (index (\<Pi>\<^sub>\<V>) v))\<^sup>+} \<union> {(Operator k (index (\<Pi>\<^sub>\<O>) op))\<^sup>+ |op. op \<in> set (\<Pi>\<^sub>\<O>) \<and> v \<in> set (delete_effects_of op)}"
eventually_Lim_ident_at: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and X :: "'a set" shows "eventually (P (Lim (at x within X) (\<lambda>x. x))) (at x within X) = eventually (P x) (at x within X)"
pathstart_join: fixes g1 :: "real \<Rightarrow> 'a"   and g2 :: "real \<Rightarrow> 'a" shows "pathstart (g1 +++ g2) = pathstart g1"
Main: fixes t :: "nat"   and n :: "nat"   and x :: "'a" assumes "t < s" shows "u (n * s + t) x \<le> Z t n x"
ideal_game_2_def: fixes x :: "'grp \<times> 'grp"   and y :: "bool"   and z :: "'e"   and A :: "(bool \<Rightarrow> 'e \<Rightarrow> (bool \<times> ('grp \<times> 'grp \<times> 'grp \<times> 'grp \<times> 'grp) \<times> 'f) spmf) \<times> (bool \<Rightarrow> 'e \<Rightarrow> 'grp \<Rightarrow> ('grp \<times> 'grp \<times> 'grp \<times> 'grp \<times> 'grp) \<times> 'f \<Rightarrow> 'd spmf)" shows "mal_def.ideal_game_2 x y z A = (let (A1, A2) = A in A1 y z \<bind> (\<lambda>(y', aux_out). mal_def.trusted_party x y' \<bind> (\<lambda>(out1, out2). A2 y' z out2 aux_out \<bind> (\<lambda>out2'. return_spmf (out1, out2')))))"
simps_6: fixes v :: "'a formula"   and va :: "'a formula" shows "is_disj (v \<^bold>\<rightarrow> va) = is_lit_plus (v \<^bold>\<rightarrow> va)"
SBWriteNonVolatile': fixes sb' :: "'c memref list"   and sb :: "'c memref list"   and a :: "nat"   and D :: "nat set"   and f :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and \<theta> :: "nat \<Rightarrow> nat option"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and "is" :: "instr list"   and m :: "nat \<Rightarrow> nat"   and ghst :: "'d \<times> 'e \<times> 'f \<times> 'g" assumes "sb' = sb @ [Write\<^sub>s\<^sub>b False a (D, f) (f \<theta>) A L R W]" shows "(Write False a (D, f) A L R W # is, \<theta>, sb, m, ghst) \<rightarrow>\<^sub>s\<^sub>b (is, \<theta>, sb', m, ghst)"
sumhr_if: fixes m :: "nat star"   and n :: "nat star"   and f :: "nat \<Rightarrow> real" shows "sumhr (m, n + 1, f) = (if n + 1 \<le> m then 0 else sumhr (m, n, f) + (*f* f) n)"
vec_dim_vec_of_poly_n_1: fixes p :: "'b poly"   and n :: "nat" shows "dim_vec (vec_of_poly_n p n) = n"
nz_impl_b_least2_le: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and x :: "nat"   and z :: "nat"   and y :: "nat" assumes "f x z \<noteq> 0" shows "b_least2 f x y \<le> z"
euler_char_def: shows "H.euler_char \<equiv> int (card (verts H)) - int (card (arcs H) div 2) + int (card H.face_cycle_sets)"
rep_eq: fixes x :: "perm"   and xa :: "('b, 'c, 'd) formula" shows "Rep_formula (x \<bullet> xa) = x \<bullet> Rep_formula xa"
update_nil_some: fixes e :: "('f, 'g, 'h) env"   and env :: "('f, 'g, 'h) env" shows "update [] (Some e) env = e"
absorb_iff1: fixes a :: "'d"   and b :: "'d" shows "(a \<le> b) = (inf a b = a)"
dyad_rec2_def: fixes u :: "'b"   and v :: "'b"   and lc :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and rc :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "real" shows "dyad_rec2 u v lc rc x = dyad_rec (\<lambda>z. (u, v)) (\<lambda>(a, b). (a, lc a b (midpoint a b))) (\<lambda>(a, b). (rc a b (midpoint a b), b)) (2 * x)"
parSym: fixes P :: "pi"   and Q :: "pi" shows "P \<parallel> Q \<sim>\<^sub>e Q \<parallel> P"
aform_inf_inner_def: fixes prec :: "nat"   and X :: "(real \<times> real pdevs) list"   and n :: "real list" shows "aform_inf_inner prec X n = (case inner_aforms' prec X (map pdevs_of_real n) of Some Xn \<Rightarrow> Inf_aform' prec (hd Xn))"
wf_projectTable: fixes n :: "nat"   and X :: "nat set \<times> 'a option list set"   and I :: "nat set" assumes "wf_atable n X" shows "wf_atable n (projectTable I X) \<and> fst (projectTable I X) = fst X \<inter> I"
labeled_saturation_lifting: fixes NL :: "('f \<times> 'l) set" assumes "saturated NL" shows "local.no_labels.saturated (fst ` NL)"
simps_28: fixes vb :: "bexp"   and vc :: "'a acom"   and vd :: "'a acom"   and ve :: "'a"   and v :: "'a acom"   and va :: "'a acom" shows "(IF vb THEN vc ELSE vd {ve} \<sqsubseteq> v;; va) = False"
mpt_I: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" assumes "sigma_finite_measure M"   and "T \<in> M \<rightarrow>\<^sub>M M"   and "\<And>A. A \<in> sets M \<Longrightarrow> emeasure M (T -` A \<inter> space M) = emeasure M A" shows "mpt M T"
le_dag_refl: fixes x :: "dag" shows "x \<le> x"
init_Accessible_dfa: shows "init Accessible.Accessible_dfa = init Accessible_dfa"
set_mult_assoc: fixes G :: "('c, 'd) monoid_scheme"   and M :: "'c set"   and H :: "'c set"   and K :: "'c set" assumes "Group.group G"   and "M \<subseteq> carrier G"   and "H \<subseteq> carrier G"   and "K \<subseteq> carrier G" shows "M <#>\<^bsub>G\<^esub> H <#>\<^bsub>G\<^esub> K = M <#>\<^bsub>G\<^esub> (H <#>\<^bsub>G\<^esub> K)"
map_le_trans: fixes m1 :: "'b \<Rightarrow> 'c option"   and m2 :: "'b \<Rightarrow> 'c option"   and m3 :: "'b \<Rightarrow> 'c option" assumes "m1 \<subseteq>\<^sub>m m2"   and "m2 \<subseteq>\<^sub>m m3" shows "m1 \<subseteq>\<^sub>m m3"
arcs_ends_conv: fixes G :: "('c, 'd) pre_digraph" shows "arcs_ends G = (\<lambda>e. (tail G e, head G e)) ` pre_digraph.arcs G"
associated_iff_same_ideal: fixes a :: "'a list"   and b :: "'a list" assumes "a \<in> carrier (K [X])"   and "b \<in> carrier (K [X])" shows "a \<sim>\<^bsub>K [X]\<^esub> b = (PIdl\<^bsub>K [X]\<^esub> a = PIdl\<^bsub>K [X]\<^esub> b)"
spanE: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "p \<in> pmdl B" obtains A :: "('t \<Rightarrow>\<^sub>0 'c) set" and q :: "('t \<Rightarrow>\<^sub>0 'c) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'c" where "finite A" and "A \<subseteq> B" and "p = (\<Sum>b\<in>A. q b \<odot> b)"
not_le: fixes x :: "'a word"   and y :: "'a word" shows "(\<not> x \<le>s y) = (y <s x)"
prime_elem_iff_irreducible: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "comm_semiring_1.prime_elem times one zero x = comm_semiring_1.irreducible times one zero x"
X_def: fixes i :: "'i" shows "X i = rind (A i)"
nrename_lredu: fixes M :: "trm"   and x :: "name"   and y :: "name"   and M' :: "trm" assumes "M[x\<turnstile>n>y] \<longrightarrow>\<^sub>l M'" shows "\<exists>M0. M0[x\<turnstile>n>y] = M' \<and> M \<longrightarrow>\<^sub>l M0"
eq: shows "(t1 * t2 - int (D * u1 * u2))\<^sup>2 - int D * (t1 * int u2 - t2 * int u1)\<^sup>2 = k\<^sup>2"
assms_3: shows "src (fst \<mu>\<nu>) = trg (snd \<mu>\<nu>)"
reaches1_reaches_iff1: fixes x :: "'b"   and y :: "'b" shows "B.reaches1 x y = (\<exists>z. B x z \<and> B.reaches z y)"
of_int_code_1: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and k :: "num" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (Int.Neg k) = uminus (numeral.numeral one plus k)"
pderivs_lang_union: fixes A :: "'a list set"   and B :: "'a list set"   and r :: "'a rexp" shows "pderivs_lang (A \<union> B) r = pderivs_lang A r \<union> pderivs_lang B r"
assms_2: shows "\<guillemotleft>\<mu> : L f \<Rightarrow>\<^sub>S f\<guillemotright>"
Disj_3I: fixes A :: "fm"   and C :: "fm"   and H :: "fm set"   and B :: "fm" assumes "insert (Neg A) (insert (Neg C) H) \<turnstile> B" shows "H \<turnstile> A OR B OR C"
action_memoryless_on: fixes f :: "'a \<Rightarrow> 'a pmf"   and s :: "'a" shows "action (memoryless_on f s) = f s"
irrlist_listassoc_cong: fixes as :: "'a list"   and bs :: "'a list" assumes "\<forall>a\<in>set as. irreducible G a"   and "as [\<sim>] bs"   and "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G" shows "\<forall>a\<in>set bs. irreducible G a"
distinct_241: fixes x101 :: "Syntax.term"   and x102 :: "Syntax.term"   and x103 :: "Syntax.term"   and x14 :: "Syntax.term" shows "Case x101 x102 x103 \<noteq> Unroll x14"
pref_profile_wfI': fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool" assumes "\<And>i. i \<in> agents' \<Longrightarrow> total_preorder_on alts' (R i)"   and "\<And>i. i \<notin> agents' \<Longrightarrow> R i = (\<lambda>_ _. False)" shows "lowered.is_pref_profile R"
makespan_mono_2: fixes T :: "nat \<Rightarrow> nat"   and x :: "nat"   and y :: "nat" assumes "T x \<le> y" shows "makespan T \<le> makespan (T(x := y))"
nm2: shows "netgmap sr n2 = (\<sigma>\<^sub>2, \<zeta>\<^sub>2)"
has_locks_iff: fixes l :: "('d \<times> nat) option"   and t :: "'d"   and n :: "nat" shows "(has_locks l t = n) = (l = None \<and> n = 0 \<or> (\<exists>n'. l = \<lfloor>(t, n')\<rfloor> \<and> Suc n' = n) \<or> (\<exists>t' n'. l = \<lfloor>(t', n')\<rfloor> \<and> t' \<noteq> t \<and> n = 0))"
has_bochner_integral_integral_eq: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and x :: "'d" assumes "has_bochner_integral M f x" shows "integral\<^sup>L M f = x"
Rep_ordinal: fixes x :: "ordinal" shows "Rep_ordinal x \<in> UNIV // ord0rel"
bij: shows "bij (\<lambda>(a, b, c). ((a, b), c))"
wf_irrefl: fixes r :: "('a \<times> 'a) set"   and a :: "'a" assumes "wf r" obtains "(a, a) \<notin> r"
assms_1: shows "2 ^ n \<le> i \<and> 2 ^ n \<le> j"
gcd_unique: fixes d :: "'a"   and a :: "'a"   and b :: "'a" shows "(d dvd a \<and> d dvd b \<and> normalize d = d \<and> (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d)) = (d = normalization_euclidean_semiring_class.gcd a b)"
induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat"   and a2 :: "nat" assumes "\<And>n k acc. (\<not> n \<le> k \<Longrightarrow> P n (Suc k) (acc * (2 * (2 * k + 1)) div (k + 2))) \<Longrightarrow> P n k acc" shows "P a0 a1 a2"
cod_simps_3: shows "\<mu>\<nu>\<pi>.cod.leg1 = \<mu>\<nu>\<pi>.leg1"
graph_Entry_axioms: shows "graph_Entry \<alpha>e \<alpha>n invar inEdges' Entry"
col: shows "multcol k a A = A * multrow_mat n k a"
finite_sysI: fixes point_set :: "'b set"   and block_collection :: "'b set multiset" assumes "incidence_system point_set block_collection"   and "finite point_set" shows "finite_incidence_system point_set block_collection"
Disj_E: fixes A :: "fm"   and H :: "fm set"   and C :: "fm"   and B :: "fm" assumes "insert A H \<turnstile> C"   and "insert B H \<turnstile> C" shows "insert (A OR B) H \<turnstile> C"
lt_set_def: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" shows "punit.lt_set F = lpp ` (F - {0})"
ide_dom: fixes f :: "(bool, bool list) concrete_category.arr" assumes "arr f" shows "ide (local.dom f)"
mult_scalar_def: fixes p :: "'a \<Rightarrow>\<^sub>0 'c" shows "(\<odot>) p = lift_poly_fun ((*) p)"
iH: shows "finprod G h F = finprod G f F \<otimes> finprod G g F"
dg_prod_2_is_arrI: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and a :: "V"   and c :: "V"   and g :: "V"   and b :: "V"   and d :: "V"   and f :: "V" assumes "digraph \<alpha> \<AA>"   and "digraph \<alpha> \<BB>"   and "g : a \<mapsto>\<^bsub>\<AA>\<^esub> c"   and "f : b \<mapsto>\<^bsub>\<BB>\<^esub> d" shows "[g, f]\<^sub>\<circ> : [a, b]\<^sub>\<circ> \<mapsto>\<^bsub>\<AA> \<times>\<^sub>D\<^sub>G \<BB>\<^esub> [c, d]\<^sub>\<circ>"
size_fib': fixes h :: "nat" shows "size (fib_tree h) = fib' h"
simulation_silent1_aux: fixes s1 :: "'s1"   and s2 :: "'s2"   and s1' :: "'s1"   and \<mu>1 :: "'s1 \<Rightarrow> 's1 \<Rightarrow> bool" assumes "s1 \<approx> s2"   and "s1 -\<tau>1\<rightarrow> s1'" shows "s1' \<approx> s2 \<and> \<mu>1\<^sup>+\<^sup>+ s1' s1 \<or> (\<exists>s2'. s2 -\<tau>2\<rightarrow>+ s2' \<and> s1' \<approx> s2')"
covector_sup_closed: fixes x :: "'b"   and y :: "'b" assumes "comp_inf.covector x"   and "comp_inf.covector y" shows "comp_inf.covector (x \<squnion> y)"
le_Err: fixes e :: "'a err"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "e \<sqsubseteq>\<^bsub>Err.le r\<^esub> Err"
that: fixes a :: "complex" assumes "\<And>w. w \<noteq> a \<Longrightarrow> infinite {x \<in> M - {z}. f x = w}" shows "thesis"
nth_rotate_conv_nth: fixes m :: "nat"   and xs :: "'b list"   and n :: "nat" assumes "m < length xs" shows "rotate n xs ! m = xs ! ((m + n) mod length xs)"
nn_integral_count_space: fixes A :: "'c set"   and f :: "'c \<Rightarrow> ennreal" assumes "finite {a \<in> A. 0 < f a}" shows "integral\<^sup>N (count_space A) f = (\<Sum>a | a \<in> A \<and> 0 < f a. f a)"
induct: fixes P :: "('a, 'b) tm \<Rightarrow> bool"   and tm :: "('a, 'b) tm" assumes "\<And>x. P (Hd x)"   and "\<And>x1a x2. \<lbrakk>P x1a; P x2\<rbrakk> \<Longrightarrow> P (App x1a x2)" shows "P tm"
simps_1: fixes head :: "'a"   and tail :: "'a iarray"   and ret0 :: "'b list"   and ret1 :: "(nat \<times> 'b) list" shows "next_subseqs1 head tail ret0 ret1 [] = (ret0, head, tail, ret1)"
simps_2: fixes \<mu> :: "'c" shows "Nml \<^bold>\<langle>\<mu>\<^bold>\<rangle> = C.arr \<mu>"
dvd_mult_cancel_right: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.idom times one plus zero minus uminus" shows "dvd.dvd times (times a c) (times b c) = (c = zero \<or> dvd.dvd times a b)"
distinct_prime_factors_nat: fixes n :: "nat" shows "distinct (prime_factors_nat n)"
assms_2: shows "constructive_security_obsf real2 ideal2 sim2 (\<I>_real2_core \<oplus>\<^sub>\<I> \<I>_real2_rest) (\<I>_ideal2_core \<oplus>\<^sub>\<I> \<I>_ideal2_rest) (\<I>_common2_core \<oplus>\<^sub>\<I> \<I>_common2_rest) (absorb \<A> (obsf_converter (fused_wiring \<odot> (converter_of_resource real1 |\<^sub>\<propto> 1\<^sub>C)))) adv2"
boundedE: fixes b :: "'b"   and c :: "'b"   and a :: "'b" assumes "b \<squnion> c \<le> a" obtains "b \<le> a" and "c \<le> a"
base: shows "q1 = 0 \<or> q2 = 0 \<or> b1 = b2"
prems_5: assumes "llast P \<in> VV p" shows "\<sigma> (llast P) = lhd P'"
pr_var_def: fixes x :: "'a \<Longrightarrow> 'b" shows "&x = x"
assms: shows "x \<in>\<^sub>\<circ> vconst_on A c \<restriction>\<^sup>l\<^sub>\<circ> B"
last_nonzero_row_def: shows "last_nonzero_row = (GREATEST m. \<not> is_zero_row_upt_k m k A)"
dvd_triv_right: fixes times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and one :: "'k"   and a :: "'k"   and b :: "'k" assumes "class.comm_monoid_mult times one" shows "dvd.dvd times a (times b a)"
power_le_imp_le_base: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and n :: "nat"   and b :: "'a" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq (power.power one times a (Suc n)) (power.power one times b (Suc n))"   and "less_eq zero b" shows "less_eq a b"
tr_par_trms_subset: fixes A' :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and A :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list"   and D :: "('lbl strand_label \<times> ('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" assumes "A' \<in> set (tr\<^sub>p\<^sub>c A D)" shows "trms\<^sub>l\<^sub>s\<^sub>t A' \<subseteq> trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<union> pair ` setops\<^sub>s\<^sub>s\<^sub>t (unlabel A) \<union> pair ` snd ` set D"
simps_2: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and f :: "'fsym"   and Tl :: "'fsym trm list" shows "F.int \<xi> (Fn f Tl) = intFF f (map (F.int \<xi>) Tl)"
lub_eq_bottom_iff: fixes Y :: "nat \<Rightarrow> 'b" assumes "chain Y" shows "((\<Squnion>i. Y i) = \<bottom>) = (\<forall>i. Y i = \<bottom>)"
combine_common_factor: fixes a :: "'c \<times> 'c \<Rightarrow> 'd"   and e :: "'c \<times> 'c \<Rightarrow> 'd"   and b :: "'c \<times> 'c \<Rightarrow> 'd"   and c :: "'c \<times> 'c \<Rightarrow> 'd" shows "a \<odot> e \<oplus> (b \<odot> e \<oplus> c) = (a \<oplus> b) \<odot> e \<oplus> c"
assms_1: shows "\<HH> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"
sets_eq_bot: fixes M :: "'c measure" shows "(sets M = {{}}) = (M = bot)"
X_def: shows "X \<equiv> fls_X_intpow df"
eq: shows "\<forall>\<^sub>F x in sequentially. contour_integral (circlepath z r) (\<lambda>u. \<Sum>k<x. (w - z) ^ k * (f u / (u - z) ^ Suc k)) = (\<Sum>k<x. contour_integral (circlepath z r) (\<lambda>u. f u / (u - z) ^ Suc k) * (w - z) ^ k)"
assms: shows "{$$} \<turnstile>\<^sub>W e : \<tau>"
fresh_5: fixes a :: "vrs"   and x2 :: "ty"   and x1 :: "ty" shows "a \<sharp> x2 \<rightarrow> x1 = (a \<sharp> x2 \<and> a \<sharp> x1)"
nth_Cons: fixes x :: "'c"   and xs :: "'c list"   and n :: "nat" shows "(x # xs) ! n = (case n of 0 \<Rightarrow> x | Suc k \<Rightarrow> xs ! k)"
cindex_poly_ubd_eventually: fixes q :: "real poly"   and p :: "real poly" shows "\<forall>\<^sub>F r in at_top. cindexE (- r) r (\<lambda>x. poly q x / poly p x) = real_of_int (cindex_poly_ubd q p)"
absorb_Some_JumpD: fixes j :: "jump"   and s :: "abopt"   and j' :: "jump" assumes "absorb j s = Some (Jump j')" shows "j' \<noteq> j"
simps_2: fixes x2 :: "'b"   and y2 :: "'b" shows "(\<^bold>\<langle>x2\<^bold>\<rangle> = \<^bold>\<langle>y2\<^bold>\<rangle>) = (x2 = y2)"
list_assn_aux_append_Cons: fixes xs :: "'b list"   and zsl :: "'c list"   and A :: "'b \<Rightarrow> 'c \<Rightarrow> assn"   and x :: "'b"   and ys :: "'b list"   and z :: "'c"   and zsr :: "'c list" assumes "length xs = length zsl" shows "list_assn A (xs @ x # ys) (zsl @ z # zsr) = list_assn A xs zsl * A x z * list_assn A ys zsr"
dg_prod_is_arrD_7: fixes \<alpha> :: "V"   and I :: "V"   and \<AA> :: "V \<Rightarrow> V"   and a :: "V"   and b :: "V"   and f :: "V"   and i :: "V" assumes "pdigraph_base \<alpha> I \<AA>"   and "f : a \<mapsto>\<^bsub>dg_prod I \<AA>\<^esub> b"   and "i \<in>\<^sub>\<circ> I" shows "f\<lparr>i\<rparr> : a\<lparr>i\<rparr> \<mapsto>\<^bsub>\<AA> i\<^esub> b\<lparr>i\<rparr>"
closure_def: fixes S :: "'d set" shows "closure S = S \<union> {x. x islimpt S}"
PhiWhilePMonotone: fixes A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and b :: "BExpr" shows "Monotone (PhiWhileP A b)"