hcju/isabelleps huggingface dataset:
l_get_dom_component_get_scdom_component_axioms: shows "l_get_dom_component_get_scdom_component get_owner_document heap_is_wellformed type_wf known_ptr known_ptrs get_scdom_component get_dom_component"
matched_bracket_param: fixes n\<^sub>0 :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n\<^sub>1 :: "'node SDG_node"   and V :: "'var"   and p :: "'pname"   and n\<^sub>2 :: "'node SDG_node"   and ns' :: "'node SDG_node list"   and n\<^sub>3 :: "'node SDG_node"   and V' :: "'var"   and n\<^sub>4 :: "'node SDG_node"   and a :: "'edge"   and a' :: "'edge" assumes "matched n\<^sub>0 ns n\<^sub>1"   and "n\<^sub>1 -p:V\<rightarrow>\<^bsub>in\<^esub> n\<^sub>2"   and "matched n\<^sub>2 ns' n\<^sub>3"   and "n\<^sub>3 -p:V'\<rightarrow>\<^bsub>out\<^esub> n\<^sub>4"   and "valid_edge a"   and "a' \<in> get_return_edges a"   and "sourcenode a = parent_node n\<^sub>1"   and "targetnode a = parent_node n\<^sub>2"   and "sourcenode a' = parent_node n\<^sub>3"   and "targetnode a' = parent_node n\<^sub>4" shows "matched n\<^sub>0 (ns @ n\<^sub>1 # ns' @ [n\<^sub>3]) n\<^sub>4"
resources_indistinguishable: fixes \<eta> :: "nat" shows "(UNIV <+> nlists UNIV (id' \<eta>) <+> UNIV) \<turnstile>\<^sub>R RES (callee_sec_channel sim \<oplus>\<^sub>O \<dagger>\<dagger>S_CHAN.send_oracle \<oplus>\<^sub>O \<dagger>\<dagger>S_CHAN.recv_oracle) (None, (), Void) \<approx> RES (\<dagger>CIPHER.AUTH.auth_oracle \<oplus>\<^sub>O key_channel_send \<eta> \<oplus>\<^sub>O key_channel_recv \<eta>) (None, Void)"
left_adjoint_preserved_by_iso': fixes f :: "('a S.E.term, 'a) S.arr"   and f' :: "('a S.E.term, 'a) S.arr" assumes "S.is_left_adjoint f"   and "S.isomorphic f f'" shows "S.is_left_adjoint f'"
strong_del_block_fin: fixes b :: "'a set" shows "finite_incidence_system (\<V> - b - b) (fin.str_del_block b)"
commute: fixes a :: "'c"   and b :: "'c" shows "a + b = b + a"
list_emb_strict_suffix: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "list_emb P xs ys"   and "strict_suffix ys zs" shows "list_emb P xs zs"
countable_acc: fixes X :: "'s set" assumes "countable X" shows "countable (K.acc `` X)"
assms_2: shows "\<forall>x. x \<in> L \<longrightarrow> f x \<in> L"
moebius_prod_not_coprime: fixes N :: "nat"   and d :: "nat" assumes "\<not> coprime N d" shows "moebius_mu (N * d) = (0::'a)"
mbuf2t_take_induct: fixes f :: "'b option list set \<Rightarrow> 'b option list set \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and buf :: "'b option list set list \<times> 'b option list set list"   and nts :: "nat list"   and z' :: "'a"   and buf' :: "'b option list set list \<times> 'b option list set list"   and nts' :: "nat list"   and i :: "nat"   and ja :: "nat"   and jb :: "nat"   and P :: "nat \<Rightarrow> 'b option list set \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> 'b option list set \<Rightarrow> bool"   and R :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and j :: "nat"   and U :: "nat \<Rightarrow> 'a \<Rightarrow> bool" assumes "mbuf2t_take f z buf nts = (z', buf', nts')"   and "wf_mbuf2 i ja jb P Q buf"   and "list_all2 R [i..<j] nts"   and "ja \<le> j"   and "jb \<le> j"   and "U i z"   and "\<And>k x y t z. \<lbrakk>i \<le> k; Suc k \<le> ja; Suc k \<le> jb; P k x; Q k y; R k t; U k z\<rbrakk> \<Longrightarrow> U (Suc k) (f x y t z)" shows "U (min ja jb) z'"
p_res_ring_0: fixes p :: "int" assumes "padic_integers p" shows "carrier (residue_ring (p ^ 0)) = {0}"
expand_superset: fixes A :: "'d set"   and g :: "'d \<Rightarrow> 'a" assumes "finite A"   and "{a. g a \<noteq> \<^bold>1} \<subseteq> A" shows "G g = F.F g A"
Proc_CFG_Call_Intra_edge_not_same_target: fixes prog :: "cmd"   and n :: "label"   and p :: "char list"   and es :: "expr list"   and rets :: "char list list"   and n' :: "label"   and n'' :: "label"   and et :: "(char list, val, char list \<times> label, char list) edge_kind" assumes "prog \<turnstile> n -CEdge (p, es, rets)\<rightarrow>\<^sub>p n'"   and "prog \<turnstile> n'' -IEdge et\<rightarrow>\<^sub>p n'" shows "False"
assms_2: shows "c_secure step out s\<^sub>0 I D"
PsiInv_alpha2: shows "\<turnstile> alpha2 \<and> $PsiInv \<longrightarrow> PsiInv$"
cSim: shows "weak_reduction_correspondence_simulation (indRelLTPO TRel) (STCal Source Target)"
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"
rel_witness_resource1: fixes A :: "'c \<Rightarrow> 'g \<Rightarrow> bool"   and A' :: "'g \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and res :: "('c, 'e) resource"   and res' :: "('d, 'f) resource" assumes "rel_resource (A OO A') B res res'"   and "left_unique A"   and "right_total A"   and "right_unique A'"   and "left_total A'" shows "rel_resource A (\<lambda>b (b', c). b = b' \<and> B b' c) res (rel_witness_resource A A' B (res, res'))"
exact_seq_epi_eq_triviality: fixes D :: "'b monoid"   and C :: "'b monoid"   and B :: "'b monoid"   and A :: "'b monoid"   and h :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'b" assumes "exact_seq ([D, C, B, A], [h, g, f])" shows "(f ` carrier A = carrier B) = trivial_homomorphism B C g"
sum_eq_1_iff: fixes A :: "'b set"   and f :: "'b \<Rightarrow> nat" assumes "finite A" shows "(sum f A = 1) = (\<exists>a\<in>A. f a = 1 \<and> (\<forall>b\<in>A. a \<noteq> b \<longrightarrow> f b = 0))"
lexLessBackjump: fixes p :: "('b \<times> bool) list"   and level :: "nat"   and a :: "('b \<times> bool) list"   and x :: "'b" assumes "p = prefixToLevel level a"   and "0 \<le> level"   and "level < currentLevel a" shows "(p @ [(x, False)], a) \<in> lexLess"
div: shows "p \<union> r2 division_of \<Union> p \<union> \<Union> (r1 - p) \<inter> \<Union> q"
mono2mono_map_option: fixes orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "'c \<Rightarrow> 'd option"   and f :: "'d \<Rightarrow> 'e" assumes "monotone orda option_ord t" shows "monotone orda option_ord (\<lambda>x. map_option f (t x))"
eq_sp_to_eq_sp_fundef: fixes F1 :: "'b \<Rightarrow> 'e fundef option"   and F2 :: "'b \<Rightarrow> 'f fundef option" assumes "Subx.sp F1 = Subx.sp F2" shows "sp_fundef F1 = sp_fundef F2"
omega_leI: fixes x :: "ordinal" assumes "\<forall>n. ordinal_of_nat n \<le> x" shows "\<omega> \<le> x"
ReadResult: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and mm :: "state \<Rightarrow> Locs \<Rightarrow> Vals"   and rs :: "state \<Rightarrow> PrIds \<Rightarrow> Vals"   and p :: "PrIds" shows "\<turnstile> Read ch mm rs p \<and> (\<forall>l. $MemInv mm l) \<longrightarrow> (rs!p)$ \<noteq> #NotAResult"
ord_option_eq_simps_2: fixes x :: "'c"   and y :: "'c option" shows "le_option (Some x) y = (y = Some x)"
fact_nonzero: fixes n :: "nat" shows "fact n \<noteq> (0::'b)"
valid_def: shows "valid \<equiv> {implem (Chan x A B M) |x A B M. M \<in> payload}"
assms: shows "\<I> \<turnstile>res res \<surd>"
children_eq2_h2: fixes ptr' :: "(_) object_ptr" shows "|h2 \<turnstile> get_child_nodes ptr'|\<^sub>r = |h3 \<turnstile> get_child_nodes ptr'|\<^sub>r"
TT_below_iff: fixes x :: "tr" shows "(TT \<sqsubseteq> x) = (x = TT)"
simps_6: fixes r :: "'b rexp" shows "nullable (Star r) = True"
step_term: fixes s :: "trm"   and t :: "trm"   and u :: "trm" assumes "s \<longlonglongrightarrow> t"   and "t \<longlonglongrightarrow>\<^sup>* u" shows "s \<longlonglongrightarrow>\<^sup>* u"
Cv2_inter_Nv1_subsetof_Upsilon2: shows "C\<^bsub>\<V>2\<^esub> \<inter> N\<^bsub>\<V>1\<^esub> \<subseteq> \<Upsilon>\<^bsub>\<Gamma>2\<^esub>"
Hermite_of_row_i_iarray_def: fixes ass :: "'a \<Rightarrow> 'a"   and res :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A :: "'a iarray iarray"   and i :: "nat" shows "Hermite_of_row_i_iarray ass res A i = (if is_zero_iarray (A !! i) then A else let j = least_non_zero_position_of_vector (A !! i); Aij = A !! i !! j; A' = mult_row_iarray A i (ass Aij div Aij) in Hermite_reduce_above_iarrays A' i i j res)"
preserves_reflects_arr: fixes f :: "'d \<times> 'c" shows "S.arr (Hom_FopxC.map f) = DopxC.arr f"
farkas_coefficients_atoms_tableau_def: fixes as :: "'b atom set"   and t :: "(nat \<times> linear_poly) list"   and C :: "(rat \<times> 'b atom) list" shows "farkas_coefficients_atoms_tableau as t C = (\<exists>p c. (\<forall>(r, a)\<in>set C. a \<in> as \<and> is_leq_ns (r *R nsc_of_atom a) \<and> r \<noteq> 0) \<and> (\<Sum>(r, a)\<leftarrow>C. lec_of_nsc (r *R nsc_of_atom a)) = Le_Constraint Leq_Rel p c \<and> c < (0::'b) \<and> (\<forall>v. v \<Turnstile>\<^sub>t t \<longrightarrow> p \<lbrace> v \<rbrace> = (0::'b)))"
build_F_correct: fixes nds :: "('a, 'b) node_scheme set"   and \<phi> :: "'a ltlr" shows "build_F nds \<phi> = {name ` A |A. \<exists>\<mu> \<eta>. A = {q \<in> nds. \<mu> U\<^sub>r \<eta> \<in> old q \<longrightarrow> \<eta> \<in> old q} \<and> \<mu> U\<^sub>r \<eta> \<in> subfrmlsr \<phi>}"
tfilter'_code_1: fixes b' :: "unit \<Rightarrow> 'b"   and P :: "'a \<Rightarrow> bool"   and b :: "'b" shows "TLList.tfilter' b' P (TNil b) = TNil b"
mgenerator2Tr3_1: fixes n :: "nat"   and j :: "nat"   and k :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and m :: "int" assumes "0 < n"   and "j \<le> n"   and "k \<le> n"   and "j = k"   and "distinct_pds K n P" shows "(\<nu>\<^bsub>K P j\<^esub>) (mprod_exp K (K_gamma k) (Kb\<^bsub>K n P\<^esub>) n\<^bsub>K\<^esub>\<^bsup>m\<^esup>) = 0"
Dom_Trg: fixes t :: "'a local.term" assumes "Arr t" shows "Dom (Trg t) = Trg t"
of_hypreal_eq_iff: fixes x :: "real star"   and y :: "real star" shows "(of_hypreal x = of_hypreal y) = (x = y)"
iFROM_inext_nth: fixes n :: "nat"   and a :: "nat" shows "[n\<dots>] \<rightarrow> a = n + a"
otrans_from_sterms: fixes \<sigma> :: "nat \<Rightarrow> 'e"   and p :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<sigma>' :: "nat \<Rightarrow> 'e"   and q :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp"   and i :: "nat" assumes "((\<sigma>, p), a, \<sigma>', q) \<in> oseqp_sos \<Gamma> i"   and "wellformed \<Gamma>" shows "\<exists>p'\<in>sterms \<Gamma> p. ((\<sigma>, p'), a, \<sigma>', q) \<in> oseqp_sos \<Gamma> i"
iT_Div_cut_less2: fixes k :: "nat"   and I :: "nat set"   and t :: "nat" assumes "0 < k" shows "(I \<oslash> k) \<down>< (t div k) = I \<down>< (t - t mod k) \<oslash> k"
cart_product_add_1_filter_eq_mirror: fixes M :: "'a multiset"   and b :: "'a set"   and N :: "'a set multiset" shows "{#m \<in># M \<times># add_mset b N. fst m \<in> snd m#} = {#m \<in># M \<times># N. fst m \<in> snd m#} + {#n \<in># M. n \<in> b#} \<times># {#b#}"
drinks_vend_invalid: fixes r :: "nat \<Rightarrow>f value option" assumes "\<nexists>n. r $ 2 = Some (Num n)" shows "possible_steps drinks 1 r STR ''vend'' [] = {||}"
UP_triangle_3: shows "S.UP ((\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>)) = S.cmp\<^sub>U\<^sub>P (trg f, f) \<cdot>\<^sub>S (S.UP \<epsilon> \<cdot>\<^sub>S S.cmp\<^sub>U\<^sub>P (f, g) \<star>\<^sub>S S.UP f) \<cdot>\<^sub>S (S.UP f \<star>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta>) \<cdot>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (f, src f))"
Int_mono: fixes A :: "'d set"   and C :: "'d set"   and B :: "'d set"   and D :: "'d set" assumes "A \<subseteq> C"   and "B \<subseteq> D" shows "A \<inter> B \<subseteq> C \<inter> D"
convert_new_thread_action_inv_1: fixes t :: "'f"   and x :: "'g"   and h :: "'h"   and f :: "'i \<Rightarrow> 'g"   and nta :: "('f, 'i, 'h) new_thread_action" shows "(NewThread t x h = convert_new_thread_action f nta) = (\<exists>x'. nta = NewThread t x' h \<and> x = f x')"
CPOST: fixes x :: "'a" shows "Q x \<Longrightarrow>\<^sub>A Q' x"
fbn_noteq: fixes a :: "('b set set, 'c) Combinators"   and aa :: "('b set set, 'c) Combinators" assumes "onlyTwoNets a"   and "onlyTwoNets aa"   and "first_bothNet a \<noteq> first_bothNet aa"   and "\<not> member DenyAll a"   and "\<not> member DenyAll aa"   and "allNetsDistinct [a, aa]" shows "first_srcNet a \<noteq> first_srcNet aa \<or> first_srcNet a \<noteq> first_destNet aa \<or> first_destNet a \<noteq> first_srcNet aa \<or> first_destNet a \<noteq> first_destNet aa"
assms_3: shows "cs\<^bsup>\<pi>\<^esup> k = cs\<^bsup>\<pi>'\<^esup> k'"
pochhammer_prod: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and n :: "nat" assumes "class.comm_semiring_1 times one plus zero" shows "comm_semiring_1.pochhammer times one plus zero a n = comm_monoid_mult.prod times one (\<lambda>i. plus a (semiring_1.of_nat one plus zero i)) {0..<n}"
pair_scale_spmf2: fixes r :: "real"   and p :: "'c spmf"   and q :: "'d spmf" assumes "r \<le> 1" shows "pair_spmf p (scale_spmf r q) = scale_spmf r (pair_spmf p q)"
normalize_sort_removed_elem_irrelevant': fixes cs :: "(String.literal \<times> String.literal) set"   and c :: "String.literal"   and s :: "String.literal set" assumes "wf_subclass_loc cs"   and "sort_ex cs (insert c s)"   and "c \<notin> normalize_sort cs (insert c s)" shows "normalize_sort cs (insert c s) = normalize_sort cs s"
resCasesF': fixes x :: "name"   and A :: "pi"   and \<alpha> :: "freeRes"   and A' :: "pi" assumes "<\<nu>x>A \<longmapsto> \<alpha> \<prec> A'" obtains P :: "pi" and \<alpha>' :: "freeRes" and P' :: "pi" and y :: "name" where "<\<nu>x>A = <\<nu>y>P" and "\<alpha> \<prec> A' = \<alpha>' \<prec> <\<nu>y>P'" and "P \<longmapsto> \<alpha>' \<prec> P'" and "y \<sharp> \<alpha>'"
fmrel_on_fsetD: fixes x :: "'e"   and S :: "'e fset"   and P :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and m :: "'e \<rightharpoonup>\<^sub>f 'f"   and n :: "'e \<rightharpoonup>\<^sub>f 'g" assumes "x |\<in>| S"   and "fmrel_on_fset S P m n" shows "rel_option P (fmlookup m x) (fmlookup n x)"
singular_boundary_eq_cycle_singleton: fixes X :: "'c topology"   and a :: "'c"   and p :: "nat"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "topspace X = {a}"   and "1 \<le> p" shows "singular_relboundary p X {} c = singular_relcycle p X {} c"
simps_3: fixes t1 :: "('c, 'd) BinomialTree"   and bq1 :: "('c, 'd) BinomialTree list"   and t2 :: "('c, 'd) BinomialTree"   and bq2 :: "('c, 'd) BinomialTree list" shows "meld (t1 # bq1) (t2 # bq2) = (if rank t1 < rank t2 then t1 # meld bq1 (t2 # bq2) else if rank t2 < rank t1 then t2 # meld (t1 # bq1) bq2 else ins (link t1 t2) (meld bq1 bq2))"
csj: shows "cs\<^bsup>path \<sigma>\<^esup> j = cs\<^bsup>path \<sigma>'\<^esup> j'"
LinLs: fixes i :: "nat"   and j :: "nat" shows "L0 i j \<in> Ls \<and> L1 i j \<in> Ls \<and> L2 i j \<in> Ls \<and> L3 i j \<in> Ls"
hyps: fixes acc :: "(bool \<times> 'b \<times> 'c) list" assumes "(ic, p, q) \<in> set (fold (\<lambda>a. case a of (ic, a) \<Rightarrow> case a of (q_in_bs, a) \<Rightarrow> case a of (p, q) \<Rightarrow> \<lambda>ps'. if \<not> ic \<and> c q_in_bs ps' p q then ps' else (ic, p, q) # ps') ps acc)" shows "(ic, p, q) \<in> set acc \<or> (\<exists>q_in_bs. (ic, q_in_bs, p, q) \<in> set ps)"
ge_plus_pos_imp_gt: fixes c :: "eint"   and b :: "eint"   and d :: "eint" assumes "c \<noteq> \<infinity>"   and "0 < b"   and "c + b \<le> d" shows "c < d"
simps_2: fixes v :: "real"   and va :: "real list" shows "is_Less (Eq v va) = False"
sup_mono: fixes sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and c :: "'e"   and b :: "'e"   and d :: "'e" assumes "class.semilattice_sup sup less_eq less"   and "less_eq a c"   and "less_eq b d" shows "less_eq (sup a b) (sup c d)"
h6: shows "(\<exists>(a', b', c')\<in>set b. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a. \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)) \<longrightarrow> (\<exists>x. (\<forall>(a, b, c)\<in>set a. a * x\<^sup>2 + b * x + c = 0) \<and> (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and> (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and> (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0))"
carrier_dim_vec: fixes v :: "'b vec"   and n :: "nat" shows "(v \<in> carrier_vec n) = (dim_vec v = n)"
divide_poly_def: fixes f :: "'b poly"   and g :: "'b poly" shows "f div g = (if g = 0 then 0 else divide_poly_main (lead_coeff g) 0 f g (degree f) (1 + length (coeffs f) - length (coeffs g)))"
sum_interval_distr: fixes m :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b"   and y :: "'b" assumes "m \<le> n" shows "sum f {m..n} \<cdot> y = (\<Sum>i = m..n. f i \<cdot> y)"
mmerge_def: fixes A :: "'a multiset"   and f :: "'a \<Rightarrow> 'm" shows "local.mmerge A f = mUnionM (image_mset f A)"
inc: shows "supp q \<subseteq> set as \<union> q \<bullet> set as"
nat_of_bool_mk_nat_vecs: fixes bs :: "bool list"   and n :: "nat" assumes "length bs = n" shows "map of_bool bs \<in> set (mk_nat_vecs n)"
sel: fixes a :: "'b" shows "the_gpv (Generative_Probabilistic_Value.Done a) = return_spmf (Pure a)"
integral_countable_UN_1: fixes s :: "nat \<Rightarrow> (real, 'a) vec set"   and f :: "(real, 'a) vec \<Rightarrow> (real, 'b) vec"   and n :: "nat" assumes "f absolutely_integrable_on \<Union> (range s)"   and "\<And>m. s m \<in> sets lebesgue" shows "f absolutely_integrable_on \<Union> (s ` {..n})"
iwlsFSb_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b, 'c, 'd, 'e) model_scheme" shows "iwlsFSb MOD \<equiv> igWlsAllDisj MOD \<and> igWlsAbsIsInBar MOD \<and> igConsIPresIGWls MOD \<and> igSubstAllIPresIGWlsAll MOD \<and> igFreshCls MOD \<and> igSubstCls MOD \<and> igAbsRen MOD"
star_subid: fixes x :: "'a" assumes "x \<le> 1'" shows "x\<^sup>\<star> = 1'"
OverriderMethodDefs_def: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and R :: "char list \<times> char list list"   and M :: "char list" shows "OverriderMethodDefs P R M \<equiv> {(Cs, mthd). \<exists>Cs' mthd'. P \<turnstile> ldc R has least M = mthd' via Cs' \<and> (Cs, mthd) \<in> MinimalMethodDefs P (mdc R) M \<and> P,mdc R \<turnstile> Cs \<sqsubseteq> snd R @\<^sub>p Cs'}"
gupd_def2: fixes r :: "oref"   and obj :: "obj"   and g :: "oref \<Rightarrow> obj option"   and l :: "lname \<Rightarrow> val option" shows "gupd(r\<mapsto>obj) (st g l) = st (g(r \<mapsto> obj)) l"
lookup_meet: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and t1 :: "('b, 'c) mapping_rbt"   and t2 :: "('b, 'c) mapping_rbt" assumes "ID ccompare \<noteq> None" shows "RBT_Mapping2.lookup (RBT_Mapping2.meet f t1 t2) = (\<lambda>k. case RBT_Mapping2.lookup t1 k of None \<Rightarrow> None | Some v1 \<Rightarrow> case RBT_Mapping2.lookup t2 k of None \<Rightarrow> None | Some v2 \<Rightarrow> Some (f k v1 v2))"
return_cont_def: fixes x :: "'d" shows "return_cont x = ContT (\<lambda>k. k x)"
dbm_lt_trans: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry"   and c :: "'c DBMEntry" assumes "a \<prec> b"   and "b \<prec> c" shows "a \<prec> c"
mbisim_mfinal: 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" shows "r1.mfinal s1 = r2.mfinal s2"
term_of_pair_pair: fixes v :: "'t" shows "term_of_pair (pp_of_term v, component_of_term v) = v"
precedesAppend: fixes a :: "'a"   and b :: "'a"   and l :: "'a list"   and l' :: "'a list" assumes "precedes a b l" shows "precedes a b (l @ l')"
commute: fixes a :: "'a word"   and b :: "'a word" shows "signed.min a b = signed.min b a"
base_eq: shows "unlift x \<leftrightarrow> unlift y"
mod_ring_divide64: shows "(mod_ring_rel64 ===> mod_ring_rel64 ===> mod_ring_rel64) (divide_p64 pp) (/)"
WT_converterD_2: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and conv :: "('a, 'b, 'c, 'd) converter"   and q :: "'a"   and r :: "'b"   and conv' :: "('a, 'b, 'c, 'd) converter" assumes "\<I>, \<I>' \<turnstile>\<^sub>C conv \<surd>"   and "q \<in> outs_\<I> \<I>"   and "(r, conv') \<in> results_gpv \<I>' (run_converter conv q)" shows "r \<in> responses_\<I> \<I> q \<and> \<I>, \<I>' \<turnstile>\<^sub>C conv' \<surd>"
remdups_upt: fixes m :: "nat"   and n :: "nat" shows "remdups [m..<n] = [m..<n]"
stepReachable_def: 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 c1 :: "('a, 'c, 'b) configuration"   and msg :: "('a, 'c) message"   and c2 :: "('a, 'c, 'b) configuration" shows "asynchronousSystem.stepReachable trans sends c1 msg c2 \<equiv> \<exists>c' c''. asynchronousSystem.reachable trans sends c1 c' \<and> asynchronousSystem.steps trans sends c' msg c'' \<and> asynchronousSystem.reachable trans sends c'' c2"
add_mono_thms_linordered_field_1: fixes i :: "'b"   and j :: "'b"   and k :: "'b"   and l :: "'b" assumes "i < j \<and> k = l" shows "i + k < j + l"
inf_disj_def: fixes n :: "nat"   and nS :: "nat \<Rightarrow> nat \<Rightarrow> 'd list" shows "inf_disj n nS \<equiv> \<forall>t i j. i < n \<and> j < n \<and> i \<noteq> j \<and> nS i t \<noteq> [] \<longrightarrow> nS j t = []"