hcju/isabelleps huggingface dataset:
path_connected_def: fixes S :: "'a set" shows "path_connected S = (\<forall>x\<in>S. \<forall>y\<in>S. \<exists>g. path g \<and> path_image g \<subseteq> S \<and> pathstart g = x \<and> pathfinish g = y)"
sat_precond_as_pair: fixes s :: "('c, 'd) fmap"   and p :: "('c, 'd) fmap"   and e :: "('c, 'd) fmap"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" shows "sat_precond_as s ((p, e) # as) = (p \<subseteq>\<^sub>f s \<and> sat_precond_as (state_succ s (p, e)) as)"
hext_refl: fixes typeof_addr :: "'b \<Rightarrow> 'c \<Rightarrow> htype option"   and h :: "'b" shows "heap_base.hext typeof_addr h h"
axioms_3: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>S\<^sub>M\<^sub>C\<^bsub>\<alpha>\<^esub> \<BB>" shows "semicategory \<alpha> \<AA>"
kpower_prop: fixes f :: "'a \<Rightarrow> 'a set"   and i :: "nat" assumes "f \<le> \<eta>" shows "kpower f i \<le> \<eta>"
scaleR_right: fixes a :: "'e \<Rightarrow>\<^sub>L 'd"   and r :: "real"   and b :: "'e" shows "blinfun_apply a (r *\<^sub>R b) = r *\<^sub>R blinfun_apply a b"
intro: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "category_with_pullbacks C"   and "category_with_terminal_object C" shows "category_with_pullbacks_and_terminal C"
simps: fixes t\<^sub>1 :: "'b"   and t\<^sub>2 :: "'b" shows "\<tau>\<^sub>c\<^sub>s\<^sub>t t\<^sub>1 * \<tau>\<^sub>c\<^sub>s\<^sub>t t\<^sub>2 = \<tau>\<^sub>c\<^sub>s\<^sub>t (t\<^sub>1 * t\<^sub>2)"
new_row_def: shows "new_row = (\<lambda>j. rec_latin_type (n - Suc m) j (R j)) ` {0..<n}"
Ide_Nmlize_Ide: fixes t :: "'c local.term" assumes "Ide t" shows "Ide \<^bold>\<lfloor>t\<^bold>\<rfloor>"
vec_cnj_sv: fixes k :: "complex"   and A :: "complex \<times> complex" shows "vec_cnj (k *\<^sub>s\<^sub>v A) = cnj k *\<^sub>s\<^sub>v vec_cnj A"
\<chi>_is_universal_3: fixes a' :: "'c"   and \<chi>' :: "'j \<Rightarrow> 'c"   and f' :: "'c" assumes "D.cone a' \<chi>'"   and "\<guillemotleft>f' : a' \<rightarrow> a\<guillemotright>"   and "D.cones_map f' \<chi> = \<chi>'" shows "f' = \<psi> (a', a) (\<Psi>.FUN a' (\<iota> \<chi>'))"
map2_rel_witness_pmf: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c pmf"   and y :: "'d pmf" assumes "rel_pmf A x y" shows "map_pmf snd (rel_witness_pmf A (x, y)) = y"
prj_simps_7: shows "C.cod f_src_src.Prj\<^sub>1\<^sub>1 = f.apex"
ereal_mult_less_0_iff: fixes a :: "ereal"   and b :: "ereal" shows "(a * b < 0) = (0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b)"
Gcd_dvd_int: fixes a :: "int"   and A :: "int set" assumes "a \<in> A" shows "Gcd A dvd a"
grid_child_without_parent: fixes p :: "(nat \<times> int) list"   and b :: "(nat \<times> int) list"   and dir :: "dir"   and d :: "nat"   and ds :: "nat set" assumes "p \<in> grid (child b dir d) ds"   and "d < length b" shows "p \<noteq> b"
Sup_\<omega>: shows "\<Union>\<^sub>\<circ> \<omega> = \<omega>"
sccs_verts_conv: shows "G.sccs_verts = verts ` G.sccs"
diff_cancel: fixes A :: "'k multiset" shows "A - A = {#}"
preserves_cod: fixes f :: "'c" assumes "C.arr f" shows "C.cod (C.map f) = C.map (C.cod f)"
sqrt_divide_self_eq: fixes x :: "real" assumes "0 \<le> x" shows "sqrt x / x = inverse (sqrt x)"
nFAND_conjuncts: fixes \<psi> :: "('a, 'b) aformula" assumes "nf_ACI \<psi>" shows "nFAND (conjuncts_list \<psi>) = \<psi>"
ICf_def: fixes i :: "nat" shows "ICf i = (\<lambda>p. IC p i)"
prec2: shows "atomic_step_precondition s2 (current s2) ipt"
intro: fixes x :: "'e \<Longrightarrow> 'd" assumes "\<And>\<sigma>. put\<^bsub>x\<^esub> \<sigma> (get\<^bsub>x\<^esub> \<sigma>) = \<sigma>" shows "wb_lens_axioms x"
assms: fixes X :: "nat \<Rightarrow> ereal" assumes "filterlim X (at_left c) sequentially" shows "(\<lambda>n. f (X n)) \<longlonglongrightarrow> y"
neg_numeral_le_floor: fixes v :: "num"   and x :: "'b" shows "(- numeral v \<le> \<lfloor>x\<rfloor>) = (- numeral v \<le> x)"
assms_2: shows "n' \<equiv> n(u := n u - {v})"
dist_not_less_zero: fixes x :: "'b"   and y :: "'b" shows "\<not> dist x y < 0"
int_conseq_seq: fixes m :: "nat"   and n :: "nat"   and l :: "nat" shows "{m..n} \<inter> {n + 1..l} = {}"
mfinal1_simulation: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "s1 \<approx>m s2"   and "r1.mfinal s1" shows "\<exists>s2'. \<tau>mRed2 s2 s2' \<and> s1 \<approx>m s2' \<and> r2.mfinal s2' \<and> shr s2' = shr s2"
preserves_ide: fixes a :: "'b \<times> 'b" assumes "B.VV.ide a" shows "B.ide (fst a \<star>\<^sub>B snd a)"
trans: fixes a :: "'a list"   and b :: "'a list"   and c :: "'a list" assumes "subseq a b"   and "subseq b c" shows "subseq a c"
rel_pmf_conj_1: fixes P :: "bool"   and Q :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b pmf"   and y :: "'c pmf" shows "rel_pmf (\<lambda>x y. P \<and> Q x y) x y = (P \<and> rel_pmf Q x y)"
fls_left_inverse_idempotent_ring1: fixes x :: "'a"   and f :: "'a fls"   and y :: "'a" assumes "x * fls_nth f (fls_subdegree f) = (1::'a)"   and "y * x = (1::'a)" shows "fls_left_inverse (fls_left_inverse f x) y = f"
inj_swap: fixes A :: "('e \<times> 'f) set" shows "inj_on prod.swap A"
hyps_1: shows "P \<turnstile> \<langle>e\<^sub>0,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow> \<langle>e\<^sub>1,s\<^sub>1,b\<^sub>1\<rangle>"
consistency: fixes ve :: "ExVar \<Rightarrow> Val option"   and e :: "Ex"   and v :: "Val"   and te :: "ExVar \<Rightarrow> Ty option"   and t :: "Ty" assumes "ve \<turnstile> e ---> v"   and "ve hastyenv te "   and "te \<turnstile> e ===> t" shows "v hasty t"
dirvec_last_polychain_of: fixes xs :: "'a list"   and p :: "'a" assumes "xs \<noteq> []" shows "dirvec (last (polychain_of p xs)) = last xs"
maps_category_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b" shows "maps_category V H \<a> \<i> src trg \<equiv> bicategory_of_spans V H \<a> \<i> src trg"
of_nat_less_of_int_iff: fixes n :: "nat"   and x :: "int" shows "(of_nat n < of_int x) = (int n < x)"
FieldI1: fixes i :: "'a"   and j :: "'a"   and R :: "('a \<times> 'a) set" assumes "(i, j) \<in> R" shows "i \<in> Field R"
butlast: shows "Es = Cs @ [D, D'] @ rev Cs''"
nat_power_eq: fixes z :: "int"   and n :: "nat" assumes "0 \<le> z" shows "nat (z ^ n) = nat z ^ n"
that: fixes a :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r"   and "a \<in>\<^sub>\<circ> A" shows "thesis"
pred_def: fixes G :: "('a, 'b) graph"   and v :: "'a" shows "pred G v \<equiv> {(v', w). (v', w, v) \<in> edges G}"
gen_enumerate_def: fixes tol :: "'h \<Rightarrow> 'i list"   and upd :: "'i \<Rightarrow> nat \<Rightarrow> 'g \<Rightarrow> 'g"   and emp :: "'g"   and S :: "'h" shows "gen_enumerate tol upd emp S \<equiv> snd (fold (\<lambda>x (k, m). (Suc k, upd x k m)) (tol S) (0, emp))"
prems_1: shows "InvariantWatchesEl (getF stateA) (getWatch1 stateA) (getWatch2 stateA)"
real_sqrt_gt_0_iff: fixes y :: "real" shows "(0 < sqrt y) = (0 < y)"
ltrm_monom_simp: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R" shows "monom P (monom P a n (deg R (monom P a n))) (deg R (monom P a n)) = monom P a n"
vsv_vimage_vsingleton: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" shows "r `\<^sub>\<circ> set {a} = set {r\<lparr>a\<rparr>}"
finiteFinestpart: fixes X :: "'c set" shows "finite (finestpart X) = finite X"
res: shows "basis_reduction_cost fs_init = (state2, c1 + c2)"
mydivcode2: fixes k :: "nat"   and D :: "nat" assumes "0 < k"   and "0 < D"   and "D < k" shows "mydiv D k = Suc 0"
silent_moves_preds_transfers_1: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as :: "'edge list"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "S,f \<turnstile> (ms,s) =as\<Rightarrow>\<^sub>\<tau> (ms',s')" shows "preds (map f as) s"
rel_mono_strong: fixes R :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and x :: "'h pmf"   and y :: "'i pmf"   and Ra :: "'h \<Rightarrow> 'i \<Rightarrow> bool" assumes "rel_pmf R x y"   and "\<And>z yb. \<lbrakk>z \<in> set_pmf x; yb \<in> set_pmf y; R z yb\<rbrakk> \<Longrightarrow> Ra z yb" shows "rel_pmf Ra x y"
3: shows "ks \<langle>(k # ks) \<langle>f\<rangle> (l # ls) \<odot> (l # ls) \<langle>g\<rangle> (m # ms)\<rangle> [m] = ks \<langle>f\<rangle> [l] \<odot> [l] \<langle>g\<rangle> [m] \<oplus> ks \<langle>f\<rangle> ls \<odot> ls \<langle>g\<rangle> [m]"
rtrancl_reflcl: fixes R :: "('c \<times> 'c) set" shows "(R\<^sup>=)\<^sup>* = R\<^sup>*"
lemma_2_5_16: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a r\<rightarrow> b \<le> (b r\<rightarrow> c) l\<rightarrow> a r\<rightarrow> c"
local_help': fixes a :: "'a list"   and r :: "'a"   and b :: "'a list"   and c :: "'a list"   and d :: "'a list" assumes "a @ r # b = c @ r # d"   and "r \<notin> set a"   and "r \<notin> set b" shows "a = c \<and> b = d"
dghm_proj_fst_HomDom: fixes \<AA> :: "V"   and \<BB> :: "V" shows "\<pi>\<^sub>D\<^sub>G\<^sub>.\<^sub>1 \<AA> \<BB>\<lparr>HomDom\<rparr> = \<AA> \<times>\<^sub>D\<^sub>G \<BB>"
cobounded1: fixes a :: "'a Word.word"   and b :: "'a Word.word" shows "a \<le>s signed.max a b"
hoare3_complete: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<Turnstile>\<^sub>3 {P} c { Q}" shows "\<turnstile>\<^sub>3 {P} c { Q}"
length_list_update: fixes xs :: "'b list"   and i :: "nat"   and x :: "'b" shows "length (xs[i := x]) = length xs"
trss_c_fmt_s: fixes s :: "'a list"   and c :: "'a list multiset"   and e :: "('b, 'c) label"   and s' :: "'a list"   and c' :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme" assumes "((s, c), e, s', c') \<in> trss fg" shows "\<exists>csp. c' = csp + c \<and> (csp = {#} \<or> (\<exists>p. e = LSpawn p \<and> csp = {#[entry fg p]#}))"
heap_is_wellformed_children_disc_nodes: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and h :: "(_) heap"   and node_ptr :: "(_) node_ptr" assumes "l_heap_is_wellformed type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_disconnected_nodes"   and "heap_is_wellformed h"   and "node_ptr |\<in>| node_ptr_kinds h"   and "\<not> (\<exists>parent\<in>fset (object_ptr_kinds h). node_ptr \<in> set |h \<turnstile> get_child_nodes parent|\<^sub>r)" shows "\<exists>document_ptr\<in>fset (document_ptr_kinds h). node_ptr \<in> set |h \<turnstile> get_disconnected_nodes document_ptr|\<^sub>r"
del_list_idem: fixes x :: "'b"   and xs :: "'b list" assumes "x \<notin> set xs" shows "del_list x xs = xs"
relpow_fun_conv: fixes a :: "'g"   and b :: "'g"   and n :: "nat"   and R :: "('g \<times> 'g) set" shows "((a, b) \<in> R ^^ n) = (\<exists>f. f 0 = a \<and> f n = b \<and> (\<forall>i<n. (f i, f (Suc i)) \<in> R))"
diff_delta_ln_upper_9_def: shows "diff_delta_ln_upper_9 \<equiv> \<lambda>x. (x - 1) ^ 9 / ((5 * x ^ 4 + 40 * x ^ 3 + 60 * x\<^sup>2 + 20 * x + 1)\<^sup>2 * x)"
alg1: fixes x :: "complex" assumes "x \<in> X" shows "algebraic x"
prognosis_upd: fixes prognosis :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<Rightarrow> Arity list \<Rightarrow> Arity \<Rightarrow> (Vars.var \<times> exp) list \<times> exp \<times> stack_elem list \<Rightarrow> Vars.var \<Rightarrow> one\<^sub>\<bottom>"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and as :: "Arity list"   and u :: "Arity"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list"   and x :: "Vars.var" assumes "CardinalityPrognosisShape prognosis" shows "prognosis ae as u (\<Gamma>, e, S) \<sqsubseteq> prognosis ae as u (\<Gamma>, e, Upd x # S)"
hull_def: shows "hull \<equiv> \<lambda>G. {x. hullp (\<lambda>x. x \<in> G) x}"
isMinim_def: fixes A :: "'a set"   and b :: "'a" shows "local.isMinim A b \<equiv> b \<in> A \<and> (\<forall>a\<in>A. (b, a) \<in> r)"
simps_4: fixes v :: "val"   and v' :: "val" shows "Abs (Some v) \<sqsubseteq> Abs (Some v') = v \<sqsubseteq> v'"
seg_start_less_end: fixes i :: "nat" assumes "i < length B" shows "seg_start i < seg_end i"
S_idx_of_def: fixes SBIP :: "'b list \<times> nat list \<times> ('b \<Rightarrow> node_state option) \<times> (nat \<times> 'b set) list"   and v :: "'b" shows "GS.S_idx_of SBIP v \<equiv> case GS.I SBIP v of Some (STACK i) \<Rightarrow> i"
outputCases': fixes a :: "name"   and b :: "name"   and P :: "pi"   and Rs :: "residual" assumes "a{b}.P \<longmapsto> Rs" obtains aa :: "name" and ba :: "name" and P :: "pi" where "a = aa \<and> b = ba \<and> Pa__ = P" and "Rs = aa[ba] \<prec> P"
not_prime_ex_mk: fixes n :: "nat" assumes "Suc 0 < n" shows "(\<exists>m k. Suc 0 < m \<and> Suc 0 < k \<and> m < n \<and> k < n \<and> n = m * k) \<or> prime n"
outs_\<I>_full: shows "outs_\<I> \<I>_full = UNIV"
H2: shows "carrier (G\<lparr>carrier := J\<rparr>) \<subseteq> J"
assign_const_rel: fixes c\<^sub>A :: "(Eg1.var, aexp, bexp) Stmt"   and x :: "Eg1.var"   and z :: "nat"   and c\<^sub>A_tail :: "(Eg1.var, aexp, bexp) Stmt"   and c\<^sub>C :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and c\<^sub>C_tail :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds\<^sub>A :: "Mode \<Rightarrow> Eg1.var set"   and mds\<^sub>C :: "Mode \<Rightarrow> var\<^sub>C set"   and mem\<^sub>A :: "Eg1.var \<Rightarrow> nat"   and mem\<^sub>C :: "var\<^sub>C \<Rightarrow> nat" assumes "c\<^sub>A = (x \<leftarrow> aexp.Const z) ;; c\<^sub>A_tail"   and "c\<^sub>C = (Eg2_var\<^sub>C_of_Eg1 x \<leftarrow> aexp\<^sub>C.Const z) ;; c\<^sub>C_tail"   and "mds\<^sub>A = mds\<^sub>A_of mds\<^sub>C"   and "mem\<^sub>A = mem\<^sub>A_of mem\<^sub>C"   and "\<forall>v\<^sub>C. v\<^sub>C \<notin> range Eg2_var\<^sub>C_of_Eg1 \<longrightarrow> v\<^sub>C \<in> mds\<^sub>C AsmNoReadOrWrite"   and "x \<notin> mds\<^sub>A GuarNoReadOrWrite"   and "x \<notin> mds\<^sub>A GuarNoWrite"   and "x \<notin> \<C>"   and "\<forall>mds\<^sub>A' mem\<^sub>A' mem\<^sub>A'' mds\<^sub>C' mem\<^sub>C' mem\<^sub>C''. mds\<^sub>A' = mds\<^sub>A_of mds\<^sub>C' \<and> mem\<^sub>A' = mem\<^sub>A_of mem\<^sub>C' \<and> mem\<^sub>A'' = mem\<^sub>A_of mem\<^sub>C'' \<and> eval_abv\<^sub>A \<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A''\<rangle>\<^sub>A \<langle>Stop ;; c\<^sub>A_tail, mds\<^sub>A', mem\<^sub>A'\<rangle>\<^sub>A \<and> eval_abv\<^sub>C \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C''\<rangle>\<^sub>C \<langle>Stop ;; c\<^sub>C_tail, mds\<^sub>C', mem\<^sub>C'\<rangle>\<^sub>C \<longrightarrow> (\<langle>Stop ;; c\<^sub>A_tail, mds\<^sub>A', mem\<^sub>A'\<rangle>\<^sub>A, \<langle>Stop ;; c\<^sub>C_tail, mds\<^sub>C', mem\<^sub>C'\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2" shows "(\<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2"
left_neutral: fixes a :: "'b list" shows "\<epsilon> \<cdot> a = a"
secure_refinement_loc_reach: fixes \<R>\<^sub>A :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) set"   and \<R> :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set"   and P :: "((('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set"   and c\<^sub>A :: "'Com\<^sub>A"   and mds\<^sub>A :: "Mode \<Rightarrow> 'Var\<^sub>A set"   and mem\<^sub>A :: "'Var\<^sub>A \<Rightarrow> 'Val"   and c\<^sub>C :: "'Com\<^sub>C"   and mds\<^sub>C :: "Mode \<Rightarrow> 'Var\<^sub>C set"   and mem\<^sub>C :: "'Var\<^sub>C \<Rightarrow> 'Val"   and c\<^sub>C' :: "'Com\<^sub>C"   and mds\<^sub>C' :: "Mode \<Rightarrow> 'Var\<^sub>C set"   and mem\<^sub>C' :: "'Var\<^sub>C \<Rightarrow> 'Val" assumes "secure_refinement \<R>\<^sub>A \<R> P"   and "(\<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C) \<in> \<R>"   and "\<langle>c\<^sub>C', mds\<^sub>C', mem\<^sub>C'\<rangle>\<^sub>C \<in> conc.loc_reach \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C" shows "\<exists>c\<^sub>A'' mds\<^sub>A'' mem\<^sub>A''. (\<langle>c\<^sub>A'', mds\<^sub>A'', mem\<^sub>A''\<rangle>\<^sub>A, \<langle>c\<^sub>C', mds\<^sub>C', mem\<^sub>C'\<rangle>\<^sub>C) \<in> \<R> \<and> \<langle>c\<^sub>A'', mds\<^sub>A'', mem\<^sub>A''\<rangle>\<^sub>A \<in> abs.loc_reach \<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A\<rangle>\<^sub>A"
last_append: fixes xs :: "'g list"   and ys :: "'g list" shows "last (xs @ ys) = (if ys = [] then last xs else last ys)"
wellformed: fixes b :: "'a set" assumes "b \<in># \<B>" shows "b \<subseteq> \<V>"
distinct_81: fixes x6 :: "nat"   and x7 :: "rat" shows "Passwd x6 \<noteq> Pri_AgrK x7"
upI_def: shows "upI = liftI (\<lambda>d l p a. upI' d l p a \<bind> (\<lambda>_. return ()))"
isSucc_pred_in: fixes i :: "'a" assumes "isSucc i" shows "(pred i, i) \<in> r"
distinct_insert: fixes x :: "'c"   and xs :: "'c list" shows "distinct (List.insert x xs) = distinct xs"
synth_analz_insert_eq: fixes X :: "msg"   and H :: "msg set"   and G :: "msg set"   and K :: "nat" assumes "X \<in> synth (analz H)"   and "H \<subseteq> G" shows "(Key K \<in> analz (insert X G)) = (Key K \<in> analz G)"
MkArr_in_hom: fixes A :: "'a \<Rightarrow> 'b"   and B :: "'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'b" assumes "A \<in> Collect (functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B))"   and "B \<in> Collect (functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B))"   and "f \<in> Collect (natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) A B)" shows "\<guillemotleft>MkArr A B f : MkIde A \<rightarrow> MkIde B\<guillemotright>"
that: fixes c :: "real" assumes "\<And>x. \<lbrakk>x\<^sub>0 \<le> x; x \<le> x\<^sub>1\<rbrakk> \<Longrightarrow> c \<le> f_approx x"   and "0 < c" shows "thesis"
real_cond_exp_gr_c: fixes f :: "'a \<Rightarrow> real"   and c :: "real" assumes "integrable M f"   and "AE x in M. c < f x" shows "AE x in M. c < real_cond_exp M Invariants f x"
Der_conc: fixes c :: "'b"   and A :: "'b list set"   and B :: "'b list set" shows "Deriv c (A @@ B) = Deriv c A @@ B \<union> (if [] \<in> A then Deriv c B else {})"
a_in_span: shows "a \<in> vec.span (R - {a})"
subseq_Cons2_iff: fixes x :: "'b"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "subseq (x # xs) (y # ys) = (if x = y then subseq xs ys else subseq (x # xs) ys)"
vec_of_list_carrier: fixes x :: "'b list" shows "vec_of_list x \<in> carrier_vec (length x)"
size_subF: fixes theta :: "vbl \<Rightarrow> vbl"   and A :: "formula" shows "size (subF theta A) = size A"
K_cfg_E: fixes cfg :: "'s cfg"   and cfg' :: "'s cfg" assumes "cfg \<in> valid_cfg"   and "cfg' \<in> set_pmf (K_cfg cfg)" shows "(state cfg, state cfg') \<in> E"
inverse_zero: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "inverse zero = zero"
eq_mrexp2_refl: fixes r :: "'c mrexp2"   and s :: "'c mrexp2" assumes "r = s" shows "eq_mrexp2 r s"
Until: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and \<phi> :: "('msaux, 'muaux) mformula"   and \<phi>' :: "Formula.formula"   and \<psi> :: "('msaux, 'muaux) mformula"   and \<psi>' :: "Formula.formula"   and args :: "Monitor.args"   and \<phi>'' :: "Formula.formula"   and I :: "\<I>"   and buf :: "event_data option list set list \<times> event_data option list set list"   and nts :: "nat list"   and aux :: "'muaux" assumes "wf_mformula \<sigma> j P V n R \<phi> \<phi>'"   and "wf_mformula \<sigma> j P V n R \<psi> \<psi>'"   and "if args_pos args then \<phi>'' = \<phi>' else \<phi>'' = formula.Neg \<phi>'"   and "safe_formula \<phi>'' = args_pos args"   and "args_ivl args = I"   and "args_n args = n"   and "args_L args = fv \<phi>'"   and "args_R args = fv \<psi>'"   and "fv \<phi>' \<subseteq> fv \<psi>'"   and "wf_mbuf2' \<sigma> P V j n R \<phi>' \<psi>' buf"   and "wf_ts \<sigma> P j \<phi>' \<psi>' nts"   and "wf_until_aux \<sigma> V R args \<phi>' \<psi>' aux (Monitor.progress \<sigma> P (formula.Until \<phi>'' I \<psi>') j)"   and "Monitor.progress \<sigma> P (formula.Until \<phi>'' I \<psi>') j + length_muaux args aux = min (Monitor.progress \<sigma> P \<phi>' j) (Monitor.progress \<sigma> P \<psi>' j)" shows "wf_mformula \<sigma> j P V n R (MUntil args \<phi> \<psi> buf nts aux) (formula.Until \<phi>'' I \<psi>')"
card_gt_0_iff: fixes A :: "'d set" shows "(0 < card A) = (A \<noteq> {} \<and> finite A)"
sup2: fixes B :: "'b"   and k :: "'b"   and b :: "'b"   and a :: "'b" assumes "B \<equiv> k \<squnion> b" shows "a \<squnion> B \<equiv> k \<squnion> (a \<squnion> b)"
prems_4: shows "fBall (frees (\<Lambda> t) |\<union>| fset_of_list \<Gamma>) (\<lambda>x. x \<le> s)"
vfsequence_vdomain: shows "\<D>\<^sub>\<circ> \<NN> = vcard \<NN>"
\<delta>_prod_precise: fixes \<delta>1 :: "('a, 'b) ta_rule set"   and t :: "'b tree"   and q1 :: "'a"   and \<delta>2 :: "('c, 'b) ta_rule set"   and q2 :: "'c" assumes "accs \<delta>1 t q1"   and "accs \<delta>2 t q2" shows "accs (\<delta>_prod \<delta>1 \<delta>2) t (q1, q2)"
wffp_cases3: fixes sl :: "('St, 'U, 'C) state list" assumes "wffp sl" obtains n :: "nat" and st :: "'St" where "0 < na__" and "sl = map Idle (replicate na__ sta__)"   | st :: "'St" and u :: "'U" and c :: "'C" and sl1 :: "('St, 'U, 'C) state list" where "sl = State stb__ ua__ ca__ # sl1a__" and "sl1a__ \<noteq> [] \<Longrightarrow> wffp sl1a__ \<and> hd sl1a__ \<in> \<delta> (State stb__ ua__ ca__)"   | n :: "nat" and st :: "'St" and u :: "'U" and c :: "'C" and sl1 :: "('St, 'U, 'C) state list" where "0 < n" and "sl = map Idle (replicate n st) @ [State (do st u c) u c] @ sl1" and "sl1 \<noteq> [] \<Longrightarrow> wffp sl1 \<and> hd sl1 \<in> \<delta> (State (do st u c) u c)"
lb: fixes v :: "'a" assumes "v \<noteq> (0::'a)" shows "\<exists>k>0. \<forall>e>0. \<exists>x\<in>S - {a}. norm (a - x) < e \<and> k * norm (a - x) \<le> \<bar>v \<bullet> (x - a)\<bar>"
x_def: fixes n :: "nat"   and a :: "'a" shows "x n a = Rep_ell2 (X n) a"
wf_residual_networkD_1: fixes x :: "'v"   and y :: "'v" assumes "wf_residual_network"   and "edge \<Delta> x y" shows "\<not> edge \<Delta> y x"
false_not_true: fixes is_true :: "'a \<Rightarrow> bool"   and is_false :: "'a \<Rightarrow> bool"   and d :: "'a" assumes "dynval is_true is_false"   and "is_false d" shows "\<not> is_true d"