hcju/isabelleps huggingface dataset:
weak_equality_def: fixes p :: "int mpoly"   and q :: "int mpoly" shows "weak_equality p q = SPEC (\<lambda>r. r \<longrightarrow> p = q)"
domain: fixes T :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "Domainp (pcr_antichain T) = (\<lambda>x. \<exists>y. rel_set T x y \<and> finite y \<and> incomparable y)"
first_isIn: fixes x :: "('d, 'e) Combinators" assumes "\<not> member DenyAll x" shows "(first_srcNet x, first_destNet x) \<in> sdnets x"
simps_1: fixes p :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "repeat 0 p = Skip"
PTrans: shows "P \<Longrightarrow>\<^sub>lu in P''\<rightarrow>a<x> \<prec> P'"
iexec_offset_aux: fixes i :: "int"   and j :: "int"   and k :: "int" shows "i + 1 - j = i - j + 1 \<and> i + j - k + 1 = i - k + j + 1"
l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and set_shadow_root :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and remove_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and remove_shadow_root_locs :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_child_nodes get_shadow_root set_shadow_root set_shadow_root_locs get_disconnected_nodes remove_shadow_root remove_shadow_root_locs \<equiv> remove_shadow_root = l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_remove_shadow_root get_child_nodes get_shadow_root set_shadow_root get_disconnected_nodes \<and> remove_shadow_root_locs = l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_remove_shadow_root_locs set_shadow_root_locs"
measurable_PiM_single: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and N :: "'b measure"   and I :: "'c set"   and M :: "'c \<Rightarrow> 'd measure" assumes "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"   and "\<And>A i. \<lbrakk>i \<in> I; A \<in> sets (M i)\<rbrakk> \<Longrightarrow> {\<omega> \<in> space N. f \<omega> i \<in> A} \<in> sets N" shows "f \<in> N \<rightarrow>\<^sub>M Pi\<^sub>M I M"
closed_vimage: fixes s :: "'c set"   and f :: "'d \<Rightarrow> 'c" assumes "closed s"   and "continuous_on UNIV f" shows "closed (f -` s)"
Q: shows "(\<box>(\<lambda>s. I s \<longrightarrow> Q s)) \<sigma>"
language_coinduct_upto_regular: fixes R :: "'a language \<Rightarrow> 'a language \<Rightarrow> bool"   and L :: "'a language"   and K :: "'a language" assumes "R L K"   and "\<And>L K. R L K \<Longrightarrow> \<oo> L = \<oo> K \<and> (\<forall>x. regular_cong R (\<dd> L x) (\<dd> K x))" shows "L = K"
S\<^sub>i\<^sub>i: fixes x :: "i"   and y :: "i" assumes "SkolemizedAxiomsSetMcL" shows "(E x \<^bold>\<and> E y \<^bold>\<leftarrow> E (x \<cdot> y)) \<^bold>\<and> (E x \<^bold>\<leftarrow> E (dom x)) \<^bold>\<and> (E y \<^bold>\<leftarrow> E (cod y))"
comparison_impl_2: fixes x :: "real"   and y :: "real" shows "(x < y) = (x \<noteq> y \<and> ge_0 (y - x))"
presWlsAbs_fromIMorAbs: fixes hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<times> 'c"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'c) Recursion.model" assumes "ipresWlsAbs hA (fromMOD MOD)" shows "presWlsAbs (fromIMorAbs hA) MOD"
abc_bcd_acd: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "[[a b c]]"   and "[[b c d]]" shows "[[a c d]]"
le_trans_antisym: fixes x :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and y :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "le_trans x y"   and "le_trans y x" shows "equiv_trans x y"
assms_3: shows "edges G1 \<subseteq> edges G2"
inverse_mod_ring_def: fixes x :: "'a mod_ring" shows "inverse x = (if x = 0 then 0 else x ^ nat (int (CARD('a) - 2)))"
restrict_apply': fixes x :: "'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "x \<in> A" shows "restrict f A x = f x"
norm_lift: fixes n :: "nat"   and xs :: "('dyn, 'b, 'fun, 'op, 'opinl) Inca.instr list"   and \<Sigma>1 :: "'c option list"   and ys :: "('dyn, 'b, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr list"   and \<Sigma>2 :: "'c option list" assumes "lift n xs \<Sigma>1 = Ok (ys, \<Sigma>2)" shows "map norm_instr ys = xs"
suminf_split_head: fixes f :: "nat \<Rightarrow> 'b" assumes "summable f" shows "(\<Sum>n. f (Suc n)) = suminf f - f 0"
cases: fixes a1 :: "'d \<Rightarrow> 'e"   and a2 :: "'d option"   and a3 :: "'d option"   and a4 :: "'e DBMEntry" assumes "dbm_entry_val a1 a2 a3 a4" obtains u :: "'d \<Rightarrow> 'e" and r :: "'d" and d :: "'e" where "a1 = u__" and "a2 = Some r__" and "a3 = None" and "a4 = Le d__" and "u__ r__ \<le> d__"   | u :: "'d \<Rightarrow> 'e" and c :: "'d" and d :: "'e" where "a1 = ua__" and "a2 = None" and "a3 = Some c__" and "a4 = Le da__" and "- ua__ c__ \<le> da__"   | u :: "'d \<Rightarrow> 'e" and r :: "'d" and d :: "'e" where "a1 = ub__" and "a2 = Some ra__" and "a3 = None" and "a4 = Lt db__" and "ub__ ra__ < db__"   | u :: "'d \<Rightarrow> 'e" and c :: "'d" and d :: "'e" where "a1 = uc__" and "a2 = None" and "a3 = Some ca__" and "a4 = Lt dc__" and "- uc__ ca__ < dc__"   | u :: "'d \<Rightarrow> 'e" and r :: "'d" and c :: "'d" and d :: "'e" where "a1 = ud__" and "a2 = Some rb__" and "a3 = Some cb__" and "a4 = Le dd__" and "ud__ rb__ - ud__ cb__ \<le> dd__"   | u :: "'d \<Rightarrow> 'e" and r :: "'d" and c :: "'d" and d :: "'e" where "a1 = u" and "a2 = Some r" and "a3 = Some c" and "a4 = Lt d" and "u r - u c < d"   | uu_ :: "'d \<Rightarrow> 'e" and uv_ :: "'d option" and uw_ :: "'d option" where "a1 = uu_" and "a2 = uv_" and "a3 = uw_" and "a4 = \<infinity>"
closed_glob_consistent_def: fixes \<R> :: "(((('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) \<times> (('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) set" shows "closed_glob_consistent \<R> = (\<forall>c\<^sub>1 mds mem\<^sub>1 c\<^sub>2 mem\<^sub>2. (\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle>, \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>) \<in> \<R> \<longrightarrow> (\<forall>A. (\<forall>x. case A x of None \<Rightarrow> True | Some (v, v') \<Rightarrow> mem\<^sub>1 x \<noteq> v \<or> mem\<^sub>2 x \<noteq> v' \<longrightarrow> \<not> var_asm_not_written mds x) \<and> (\<forall>x. dma mem\<^sub>1 [\<parallel>\<^sub>1 A] x \<noteq> dma mem\<^sub>1 x \<longrightarrow> \<not> var_asm_not_written mds x) \<and> (\<forall>x. dma mem\<^sub>1 [\<parallel>\<^sub>1 A] x = Low \<and> (x \<notin> mds AsmNoReadOrWrite \<or> x \<in> \<C>) \<longrightarrow> mem\<^sub>1 [\<parallel>\<^sub>1 A] x = mem\<^sub>2 [\<parallel>\<^sub>2 A] x) \<longrightarrow> (\<langle>c\<^sub>1, mds, mem\<^sub>1 [\<parallel>\<^sub>1 A]\<rangle>, \<langle>c\<^sub>2, mds, mem\<^sub>2 [\<parallel>\<^sub>2 A]\<rangle>) \<in> \<R>))"
hintikka_model: fixes H :: "('b, 'c) form set"   and p :: "('b, 'c) form"   and e :: "nat \<Rightarrow> 'b hterm" assumes "hintikka H" shows "(p \<in> H \<longrightarrow> closed 0 p \<longrightarrow> eval e HApp (\<lambda>a ts. Pred a (terms_of_hterms ts) \<in> H) p) \<and> (Neg p \<in> H \<longrightarrow> closed 0 p \<longrightarrow> eval e HApp (\<lambda>a ts. Pred a (terms_of_hterms ts) \<in> H) (Neg p))"
arr_fromArr: fixes x :: "'arr" assumes "x \<in> Arr" shows "arr (fromArr x)"
iii: shows "vc C (\<lambda>s. I s + 1)"
r2f_curry: fixes R :: "('a \<times> 'b) set" shows "\<F> R = Collect \<circ> (curry \<circ> s2p) R"
smult_vec_def: fixes a :: "'a"   and v :: "'a Matrix.vec" shows "a \<cdot>\<^sub>v v \<equiv> Matrix.vec (dim_vec v) (\<lambda>i. a * v $ i)"
set_integral_Un: fixes A :: "'c set"   and B :: "'c set"   and M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "A \<inter> B = {}"   and "set_integrable M A f"   and "set_integrable M B f" shows "set_lebesgue_integral M (A \<union> B) f = set_lebesgue_integral M A f + set_lebesgue_integral M B f"
lincomb_in_span: fixes X :: "'b vec set"   and n :: "nat"   and a :: "'b vec \<Rightarrow> 'b" assumes "X \<subseteq> carrier_vec n" shows "module.lincomb (module_vec TYPE('b) n) a X \<in> LinearCombinations.module.span class_ring (module_vec TYPE('b) n) X"
analytic_newman_variant: obtains c :: "complex" and a :: "complex" where "(\<lambda>z. if z = 1 then a else newman z + deriv zeta z + c * zeta z) analytic_on {s. 1 \<le> Re s}"
starset_n_def: fixes As :: "nat \<Rightarrow> 'a set" shows "*sn* As = Iset (star_n As)"
Int_Collect: fixes x :: "'f"   and A :: "'f set"   and P :: "'f \<Rightarrow> bool" shows "(x \<in> A \<inter> {x. P x}) = (x \<in> A \<and> P x)"
power_commutes: fixes a :: "'a"   and n :: "nat" shows "a ^ n * a = a * a ^ n"
simps: fixes gpv :: "('c, 'call, 'ret) gpv"   and s :: "'s" shows "local.inline1 gpv s = the_gpv gpv \<bind> case_generat (\<lambda>x. return_spmf (Inl (x, s))) (\<lambda>out rpv. the_gpv (callee s out) \<bind> case_generat (\<lambda>(x, y). local.inline1 (rpv x) y) (\<lambda>out rpv'. return_spmf (Inr (out, rpv', rpv))))"
sp_generalize2: fixes F :: "'n \<Rightarrow> 'o fundef option"   and xs :: "('c, 'p, 'n, 'e, 'f, 'g, 'h, 'i) instr list"   and n :: "nat"   and \<Sigma>' :: "Unboxed.type option list" assumes "sp F xs (replicate n None) = Ok \<Sigma>'" shows "sp F (map generalize_instr xs) (replicate n None) = Ok (map Map.empty \<Sigma>')"
rangeE: fixes b :: "'d"   and f :: "'e \<Rightarrow> 'd" assumes "b \<in> range f" obtains x :: "'e" where "b = f x"
r: shows "(flow0 x \<circ> s) \<longlonglongrightarrow> flow0 x t"
parIdRight: fixes Rel :: "(ccs \<times> ccs) set"   and P :: "ccs" assumes "\<And>Q. (Q, Q \<parallel> \<zero>) \<in> Rel" shows "P \<leadsto>[Rel] P \<parallel> \<zero>"
incl: fixes h :: "'a"   and H :: "'a set"   and R :: "('a, 'b) ring_scheme" assumes "h \<in> H" shows "h \<in> generate_field R H"
simps: fixes KP :: "('a \<times> 'b) set" shows "treap_of KP = (if infinite KP \<or> KP = {} then \<langle>\<rangle> else let m = arg_min_on snd KP; L = {p \<in> KP. fst p < fst m}; R = {p \<in> KP. fst m < fst p} in \<langle>treap_of L, m, treap_of R\<rangle>)"
small_nstep_csmall_nstep_equiv: fixes P :: "'prog"   and \<sigma> :: "'state"   and n :: "nat" shows "small_nstep P \<sigma> n = {\<sigma>''. \<exists>coll. (\<sigma>'', coll) \<in> csmall_nstep P \<sigma> n}"
vconn_implies_hpar_2: fixes \<mu> :: "('a S.E.term, 'a) S.arr"   and f :: "('a S.E.term, 'a) S.arr"   and g :: "('a S.E.term, 'a) S.arr" assumes "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>S g\<guillemotright>" shows "S.trg \<mu> = S.trg f"
simp_el_indecomp_3: fixes b :: "'b list"   and G :: "'b list set" assumes " b \<in>B G " shows "\<forall>u v. u \<noteq> \<epsilon> \<and> u \<in> \<langle>G\<rangle> \<and> v \<noteq> \<epsilon> \<and> v \<in> \<langle>G\<rangle> \<longrightarrow> u \<cdot> v \<noteq> b"
axioms_1: fixes R :: "('c, 'd) ring_scheme" assumes "cring R" shows "ring R"
totatives_le: fixes x :: "nat"   and n :: "nat" assumes "x \<in> totatives n" shows "x \<le> n"
diff_zero: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and a :: "'c" assumes "class.cancel_comm_monoid_add plus minus zero" shows "minus a zero = a"
meet_iso: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "x \<cdot> z \<le> y \<cdot> z"
SubstFormP_Mem: fixes v :: "tm"   and i :: "tm"   and x :: "tm"   and x' :: "tm"   and y :: "tm"   and y' :: "tm" shows "{SubstTermP v i x x', SubstTermP v i y y'} \<turnstile> SubstFormP v i (Q_Mem x y) (Q_Mem x' y')"
Inf_greatest: 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 z :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq z x" shows "less_eq z (Inf A)"
ACI_norm_lderivs_Pr: fixes r :: "'b rexp" shows "ACI_norm ` {lderivs xs (Pr r) |xs. True} \<subseteq> {Pr Zero} \<union> {Pr \<guillemotleft>s\<guillemotright> |s. toplevel_summands s \<subseteq> (\<Union>xs. toplevel_summands \<guillemotleft>lderivs xs r\<guillemotright>)}"
oth_class_taut_9_a: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<phi> \<^bold>& \<psi> \<^bold>\<rightarrow> \<phi> in v]"
intro: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "\<And>gt ys xs y x. \<lbrakk>\<forall>z y x. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x; \<forall>y x. gt y x \<or> gt x y \<or> y = x; length ys = length xs; ext gt (y # ys) (x # xs)\<rbrakk> \<Longrightarrow> gt y x \<or> ext gt ys xs" shows "ext_hd_or_tl_axioms ext"
compact_eq_openin_cover: fixes S :: "'b set" shows "compact S = (\<forall>C. (\<forall>c\<in>C. openin (top_of_set S) c) \<and> S \<subseteq> \<Union> C \<longrightarrow> (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union> D))"
Derives1_def: fixes u :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and v :: "symbol list" shows "Derives1 u i r v = (\<exists>x y N \<alpha>. u = x @ [N] @ y \<and> v = x @ \<alpha> @ y \<and> is_sentence x \<and> is_sentence y \<and> (N, \<alpha>) \<in> \<RR> \<and> r = (N, \<alpha>) \<and> i = length x)"
insort_insert_triv: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" assumes "class.linorder less_eq less"   and "x \<in> set xs" shows "linorder.insort_insert_key less_eq (\<lambda>x. x) x xs = xs"
sa_def: fixes G :: "('a, 'b, 'c) sa_rec_scheme" shows "sa G \<equiv> graph G"
sha1len: fixes m :: "bit list" shows "length (sha1 m) = 160"
intro: fixes I :: "'c set"   and R :: "('c, 'd) ring_scheme"   and f :: "'c \<Rightarrow> 'e" assumes "ideal I R"   and "weak_ring_morphism_axioms f I R" shows "weak_ring_morphism f I R"
strips_parallel_plan_to_sas_plus_parallel_plan_def: fixes \<Pi> :: "('c, 'd) sas_plus_problem"   and \<pi> :: "('c \<times> 'd) strips_operator list list" shows "\<phi>\<^sub>P\<inverse> \<Pi> \<pi> \<equiv> map (map (strips_op_to_sasp \<Pi>)) \<pi>"
\<omega>_limit_set_in_compact_compact: fixes K :: "'a set"   and x :: "'a" assumes "compact K"   and "K \<subseteq> X"   and "x \<in> X"   and "rev.trapped_forward x K" shows "compact (rev.\<omega>_limit_set x)"
u: shows "u \<in> \<X> k \<and> (\<exists>p\<in>by_length k P. admissible (p @ [u]))"
some_the_fst_netgmap: fixes i :: "nat"   and s :: "'c net_state"   and sr :: "'c \<Rightarrow> 'd \<times> 'e" assumes "i \<in> net_ips s" shows "Some (the (fst (netgmap sr s) i)) = fst (netgmap sr s) i"
disc_24: fixes x4 :: "'a" shows "\<not> isRecvMarker (Snapshot x4)"
dvd_dvd_smult: fixes a :: "'a"   and b :: "'a"   and f :: "'a poly"   and g :: "'a poly" assumes "a dvd b"   and "f dvd g" shows "smult a f dvd smult b g"
difference_character: fixes x :: "i"   and y :: "i" assumes "\<exists>w. P w x \<and> \<not> O w y" shows "\<forall>w. P w (x \<ominus> y) = (P w x \<and> \<not> O w y)"
det_permute_rows: fixes p :: "'a \<Rightarrow> 'a"   and A :: "(('b, 'a) vec, 'a) vec" assumes "p permutes UNIV" shows "det (\<chi>i. A $ p i) = of_int (sign p) * det A"
ls'list_def: shows "ls'list = (enumAppend nmax ^^ n) (enumBase nmax)"
binrelchain_remdup_adj: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and x :: "'b"   and ys :: "'b list" assumes "binrelchain P (xs @ [x, x] @ ys)" shows "binrelchain P (xs @ x # ys)"
real_integrable_def: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> real" shows "integrable M f = (f \<in> borel_measurable M \<and> \<integral>\<^sup>+ x. ennreal (f x) \<partial>M \<noteq> \<infinity> \<and> \<integral>\<^sup>+ x. ennreal (- f x) \<partial>M \<noteq> \<infinity>)"
1: shows "\<forall>j\<in>{0..<i}. \<phi> (map (sdrop j) \<pi>l)"
assms_5: fixes l :: "nat list"   and i :: "nat" assumes "l \<in> set subsets"   and "i \<in> set l" shows "i < length qs"
coeff_pCons_Suc: fixes a :: "'c"   and p :: "'c poly"   and n :: "nat" shows "coeff (pCons a p) (Suc n) = coeff p n"
isin_prefix_trie: fixes ps :: "bool list"   and t :: "trie"   and ks :: "bool list" shows "isin (prefix_trie ps t) ks = (ps = take (length ps) ks \<and> isin t (drop (length ps) ks))"
in_KeysE: fixes s :: "'b"   and F :: "('b \<Rightarrow>\<^sub>0 'c) set" assumes "s \<in> Keys F" obtains f :: "'b \<Rightarrow>\<^sub>0 'c" where "s \<in> keys f" and "f \<in> F"
emsapss_decode_help7_def: fixes mHash :: "bit list"   and DB_zero :: "bit list"   and H :: "bit list"   and emBits :: "nat" shows "emsapss_decode_help7 mHash DB_zero H emBits = (if take (roundup emBits 8 * 8 - length mHash - sLen - 16) DB_zero \<noteq> bv_prepend (roundup emBits 8 * 8 - length mHash - sLen - 16) \<zero> [] \<or> take 8 (drop (roundup emBits 8 * 8 - length mHash - sLen - 16) DB_zero) \<noteq> [\<zero>, \<zero>, \<zero>, \<zero>, \<zero>, \<zero>, \<zero>, \<one>] then False else emsapss_decode_help8 mHash DB_zero H)"
simps_2: fixes I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and u :: "'b"   and x :: "'a"   and xs :: "'a list" shows "ipurge_tr_rev I D u (x # xs) = (if D x \<in> sources I D u (x # xs) then x # ipurge_tr_rev I D u xs else ipurge_tr_rev I D u xs)"
reflexive: fixes p :: "'state" shows "p \<in> reachable p"
less_def: shows "less = (\<lambda>x y. (x, y) \<in> R \<and> (y, x) \<notin> R)"
hyps_8: assumes "hom_decomp (fst (shift_list (h, U) x qs, B - {(h, U)}))" shows "hom_decomp (fst (shift1 (shift_list (h, U) x qs, B - {(h, U)})))"
ID_restricted_store_subset_store_1: fixes \<sigma> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and l :: "'b" shows "RID\<^sub>S (\<sigma>(l := None)) \<subseteq> RID\<^sub>S \<sigma>"
alignUp_def2: fixes a :: "'b word"   and sz :: "nat" shows "alignUp a sz = a + 2 ^ sz - 1 AND NOT (mask sz)"
proots_ball_card_code1: fixes p :: "complex poly"   and z0 :: "complex"   and r :: "real" shows "proots_ball_card p z0 r = (if r \<le> 0 \<or> p = 0 then 0 else proots_upper_card (fcompose (p \<circ>\<^sub>p [:z0, complex_of_real r:]) [:\<i>, - 1:] [:\<i>, 1:]))"
r_tau_def: shows "r_tau \<equiv> Cn 2 r_nth [Cn 2 r_pdec1 [Cn 2 r_state [r_dettime, recf.Id 2 0]], recf.Id 2 1]"
low_mds_eq_from_conc_to_abs: fixes mds :: "Mode \<Rightarrow> var\<^sub>C set"   and mem :: "var\<^sub>C \<Rightarrow> nat"   and mem' :: "var\<^sub>C \<Rightarrow> nat" assumes "conc.low_mds_eq mds mem mem'" shows "A.low_mds_eq (mds\<^sub>A_of mds) (mem\<^sub>A_of mem) (mem\<^sub>A_of mem')"
truncate_down_less_zero_iff: fixes p :: "nat"   and x :: "real" shows "(truncate_down p x < 0) = (x < 0)"
continuous_levelset_openin_cases: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and a :: "'d" assumes "connected s"   and "continuous_on s f"   and "openin (top_of_set s) {x \<in> s. f x = a}" shows "(\<forall>x\<in>s. f x \<noteq> a) \<or> (\<forall>x\<in>s. f x = a)"
clear_Bit_eq: fixes w :: "'c word"   and n :: "nat" shows "clearBit w n = unset_bit n w"
predicate2I: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>x y. P x y \<Longrightarrow> Q x y" shows "P \<le> Q"
case_one_imp_gap_unbounded: fixes i :: "nat"   and n :: "nat" assumes "case_one i" shows "\<exists>t. n < gap i t - 1"
Restrp_semiattractive: shows "semiattractive UNIV ((\<sqsubseteq>) \<restriction> A \<restriction> UNIV)"
cases: fixes P :: "(char list list \<times> char list exp) prog"   and a1 :: "char list exp list"   and a2 :: "heap \<times> sfields \<times> sheap"   and a3 :: "char list exp list"   and a4 :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>a1,a2\<rangle> [\<Rightarrow>] \<langle>a3,a4\<rangle>" obtains s :: "heap \<times> sfields \<times> sheap" where "a1 = []" and "a2 = s" and "a3 = []" and "a4 = s"   | e :: "char list exp" and s\<^sub>0 :: "heap \<times> sfields \<times> sheap" and v :: "val" and s\<^sub>1 :: "heap \<times> sfields \<times> sheap" and es :: "char list exp list" and es' :: "char list exp list" and s\<^sub>2 :: "heap \<times> sfields \<times> sheap" where "a1 = ea__ # es__" and "a2 = s\<^sub>0__" and "a3 = Val v # es'" and "a4 = s\<^sub>2" and "P \<turnstile> \<langle>ea__,s\<^sub>0__\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1__\<rangle>" and "P \<turnstile> \<langle>es__,s\<^sub>1__\<rangle> [\<Rightarrow>] \<langle>es',s\<^sub>2\<rangle>"   | e :: "char list exp" and s\<^sub>0 :: "heap \<times> sfields \<times> sheap" and e' :: "char list exp" and s\<^sub>1 :: "heap \<times> sfields \<times> sheap" and es :: "char list exp list" where "a1 = e # es" and "a2 = s\<^sub>0" and "a3 = throw e' # es" and "a4 = s\<^sub>1" and "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
pref_not_idem: fixes z :: "'b list"   and x :: "'b list"   and k :: "nat" assumes "z \<noteq> \<epsilon>"   and "z \<noteq> x" shows "z \<cdot> x \<^sup>@ k \<noteq> x"
strips_equivalent_to_sas_plus: fixes \<Psi> :: "('a, 'b) sas_plus_problem"   and \<psi> :: "('a, 'b) sas_plus_operator list list" assumes "is_valid_problem_sas_plus \<Psi>"   and "SAS_Plus_Semantics.is_parallel_solution_for_problem \<Psi> \<psi>" shows "STRIPS_Semantics.is_parallel_solution_for_problem (\<phi> \<Psi> ) (\<phi>\<^sub>P \<Psi> \<psi>)"
simps_3: fixes a :: "MIR.num" shows "bound0 (Lt a) = numbound0 a"
assms: shows "(c', ctxts) \<in> decompose_com c"
ag_r_inv1: fixes A :: "('d, 'e) aGroup_scheme"   and x :: "'d" assumes "aGroup A"   and "x \<in> carrier A" shows "x \<plusminus>\<^bsub>A\<^esub> -\<^sub>a\<^bsub>A\<^esub> x = \<zero>\<^bsub>A\<^esub>"
mat_eq_is_eq_if_eq_forall_j: fixes v :: "i"   and F :: "\<Pi>\<^sub>1"   and G :: "\<Pi>\<^sub>1" assumes "[\<^bold>\<forall>x. \<^bold>\<box>(\<lparr>F,x\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>G,x\<^sup>P\<rparr>) in v]" shows "[F \<^bold>= G in v] = (\<forall>s. s \<noteq> dj \<longrightarrow> (\<forall>x. eval\<Pi>\<^sub>1 F x s = eval\<Pi>\<^sub>1 G x s))"
valid_result_mmsaux_unfolded: fixes args :: "Monitor.args"   and cur :: "nat"   and t :: "nat"   and gc :: "nat"   and maskL :: "bool list"   and maskR :: "bool list"   and data_prev :: "(nat \<times> 'b option list set) queue"   and data_in :: "(nat \<times> 'b option list set) queue"   and tuple_in :: "('b option list, nat) mapping"   and tuple_since :: "('b option list, nat) mapping"   and auxlist :: "(nat \<times> 'b option list set) list" assumes "valid_mmsaux args cur (t, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since) auxlist" shows "result_mmsaux args (t, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since) = foldr (\<union>) (concat (map (\<lambda>(t, rel). if left (args_ivl args) \<le> cur - t then [rel] else []) auxlist)) {}"
resets_mono_0': fixes xs :: "('c \<Rightarrow> real) stream"   and i :: "nat"   and x :: "'c"   and c :: "'c" assumes "PTA_Reachability.pairwise eq_elapsed xs"   and "stream_all (\<lambda>u. u \<in> V) xs"   and "stream_trans xs"   and "\<forall>j\<le>i. zero x ([xs !! j]\<^sub>\<R>)"   and "x \<in> \<X>"   and "c \<in> \<X>" shows "(xs !! i) c = (xs !! 0) c \<or> (xs !! i) c = 0"
subset_def: fixes A :: "ZF"   and B :: "ZF" shows "A |\<subseteq>| B \<equiv> \<forall>x. x |\<in>| A \<longrightarrow> x |\<in>| B"