hcju/isabelleps huggingface dataset:
set_relD2: fixes s :: "'a set"   and s' :: "'b set"   and R :: "('a \<times> 'b) set"   and x' :: "'b" assumes "(s, s') \<in> \<langle>R\<rangle>set_rel"   and "x' \<in> s'" shows "\<exists>x\<in>s. (x, x') \<in> R"
plays_from_strategy_lengths: fixes \<sigma> :: "'c list \<Rightarrow> 'c"   and p :: "'c list"   and i :: "nat" shows "length (strategy_progression \<sigma> p $ i) = i"
partial_division_extend: fixes p :: "'a set set"   and S :: "'a set"   and q :: "'a set set"   and T :: "'a set" assumes "p division_of S"   and "q division_of T"   and "S \<subseteq> T" obtains r :: "'a set set" where "p \<subseteq> r" and "r division_of T"
mult_1s_ring_1_1: fixes b :: "'b" shows "- Numeral1 * b = - b"
equalOn_UnD: fixes A :: "'a set"   and B :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "equalOn (A \<union> B) f g" shows "equalOn A f g \<and> equalOn B f g"
assms_2: shows "is_tangle_diagram (w2 * ws2)"
assms_1: shows "insert ((i \<leftrightarrow> i') \<bullet> A) H \<turnstile> B"
rel_inject_2: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x21 :: "'d"   and x22 :: "'d list"   and y21 :: "'e"   and y22 :: "'e list" shows "list_all2 R (x21 # x22) (y21 # y22) = (R x21 y21 \<and> list_all2 R x22 y22)"
slim_simps_1: shows "pverts slim.slim = slim.slim_verts"
field_cont_on_gamma2_image: shows "continuous_on {p. \<exists>x. x \<in> cbox a b \<and> p = (g1 x, x)} (\<lambda>a. F a \<bullet> j)"
normal_restrict_supergroup: fixes S :: "'a set"   and N :: "'a set" assumes "subgroup S G"   and "N \<lhd> G"   and "N \<subseteq> S" shows "N \<lhd> G\<lparr>carrier := S\<rparr>"
simps_10: fixes x41 :: "'d list"   and x42 :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list"   and x1 :: "('c, 'd) Term.term" shows "\<forall>x41\<langle>\<or>\<noteq>: x42\<rangle>\<^sub>s\<^sub>t \<noteq> send\<langle>x1\<rangle>\<^sub>s\<^sub>t"
residual_countable_network: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" assumes "res.wf_residual_network"   and "flow (residual_network f) f" shows "countable_network (res.residual_network f)"
greaterThanLessThan_empty_iff2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.dense_linorder less_eq less" shows "({} = ord.greaterThanLessThan less a b) = less_eq b a"
a: shows "\<forall>u. v \<prec>\<^sub>t u \<longrightarrow> lookup p u = lookup q u"
create_character_data_impl: shows "create_character_data = local.a_create_character_data"
polymul: fixes bs :: "'a list"   and p :: "poly"   and q :: "poly" shows "\<lparr>p *\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup> = \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> * \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
add: fixes x :: "'a"   and b1 :: "'b"   and b2 :: "'b" shows "f' y x (b1 + b2) = f' y x b1 + f' y x b2"
FlexRayArch_nStore: fixes n :: "nat"   and nReturn :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nC :: "nat \<Rightarrow> Config"   and nStore :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nGet :: "nat \<Rightarrow> nat \<Rightarrow> nat list"   and i :: "nat" assumes "FlexRayArch n nReturn nC nStore nGet"   and "\<forall>i<n. msg (Suc 0) (nReturn i)"   and "DisjointSchedules n nC"   and "IdenticCycleLength n nC"   and "i < n" shows "msg (Suc 0) (nStore i)"
assms: shows "low_equal s1 s2 \<and> fst (memory_read 10 addra s1) = fst (memory_read 10 addra s2)"
assms: shows "invalidate rt dests dip = Some (dsn, dsk, flag, hops, nhip)"
prime_gt_0_nat: fixes p :: "nat" assumes "prime p" shows "0 < p"
C_eq_if_mr_eq2: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and r :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and b :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "applied_rule_rev Cp x a = \<lfloor>r\<rfloor>"   and "applied_rule_rev Cp x b = \<lfloor>r\<rfloor>"   and "a \<noteq> []"   and "b \<noteq> []" shows "Cp (list2FWpolicy a) x = Cp (list2FWpolicy b) x"
adjoint_scale: fixes a :: "'a"   and A :: "'a mat" shows "adjoint (a \<cdot>\<^sub>m A) = conjugate a \<cdot>\<^sub>m adjoint A"
nn_integral_distr: fixes T :: "'d \<Rightarrow> 'e"   and M :: "'d measure"   and M' :: "'e measure"   and f :: "'e \<Rightarrow> ennreal" assumes "T \<in> M \<rightarrow>\<^sub>M M'"   and "f \<in> borel_measurable (distr M M' T)" shows "integral\<^sup>N (distr M M' T) f = \<integral>\<^sup>+ x. f (T x) \<partial>M"
WTrtFAccNT: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and F :: "char list"   and D :: "char list"   and T :: "ty" assumes "P,E,h,sh \<turnstile> e : NT" shows "P,E,h,sh \<turnstile> e\<bullet>F{D} : T"
distinct_1: shows "X4.Xd1 \<noteq> X4.Xd1'"
sngr_assn_def: fixes r :: "'b ref"   and x :: "'b" shows "r \<mapsto>\<^sub>r x = Abs_assn (Assn (\<lambda>h. Ref.get (heapOf h) r = x \<and> addrOf h = {addr_of_ref r} \<and> addr_of_ref r < lim (heapOf h)))"
find_slot_def: shows "find_slot = (\<lambda>open_flag slotable. Heap_Error_Monad.bind (get_parent slotable) (case_option (return None) (\<lambda>parent. if is_element_ptr_kind\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r parent then Heap_Error_Monad.bind (get_shadow_root (the (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r parent))) (case_option (return None) (\<lambda>shadow_root_ptr. Heap_Error_Monad.bind (get_mode shadow_root_ptr) (\<lambda>shadow_root_mode. if open_flag \<and> shadow_root_mode \<noteq> Open then return None else first_in_tree_order (cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr) (\<lambda>ptr. if is_element_ptr_kind\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr then Heap_Error_Monad.bind (get_tag_name (the (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr))) (\<lambda>tag. Heap_Error_Monad.bind (get_attribute (the (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)) ''name'') (\<lambda>name_attr. Heap_Error_Monad.bind (if is_element_ptr_kind\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r slotable then get_attribute (the (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r slotable)) ''slot'' else return None) (\<lambda>slotable_name_attr. if tag = ''slot'' \<and> (name_attr = slotable_name_attr \<or> name_attr = None \<and> slotable_name_attr = Some [] \<or> name_attr = Some [] \<and> slotable_name_attr = None) then return (Some (the (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr))) else return None))) else return None)))) else return None)))"
not_negligible_1: shows "\<not> negligible (\<lambda>_. 1)"
I: shows "I \<subseteq> {0..<dim_row (P * A * Q)}"
ltake_0: fixes xs :: "'c llist" shows "ltake 0 xs = LNil"
cring_fieldI: fixes R :: "('e, 'f) ring_scheme" assumes "cring R"   and "Units R = carrier R - {\<zero>\<^bsub>R\<^esub>}" shows "field R"
App_NF_D: fixes n :: "nat"   and ts :: "dB list" assumes "NF (dB.Var n \<degree>\<degree> ts)" shows "listall NF ts"
mes: fixes n :: "nat" shows "F n \<in> borel_measurable M"
map_poly_add: fixes h :: "'d \<Rightarrow> 'c"   and p :: "'d poly"   and q :: "'d poly" assumes "h (0::'d) = (0::'c)"   and "\<forall>p q. h (p + q) = h p + h q" shows "map_poly h (p + q) = map_poly h p + map_poly h q"
rBang: fixes Rs :: "('a, 'b, 'c) residual"   and C :: "'d" assumes "\<Psi> \<rhd> P \<parallel> !P \<longmapsto> Rs"   and "\<And>C. Prop C \<Psi> (P \<parallel> !P) Rs"   and "guarded P" shows "Prop C \<Psi> (!P) Rs"
gHom_def: fixes G :: "('e, 'g) Group_scheme"   and F :: "('f, 'h) Group_scheme" shows "gHom G F = {f. (f \<in> extensional (carrier G) \<and> f \<in> carrier G \<rightarrow> carrier F) \<and> (\<forall>x\<in>carrier G. \<forall>y\<in>carrier G. f (x \<cdot>\<^bsub>G\<^esub> y) = f x \<cdot>\<^bsub>F\<^esub> f y)}"
adjoint_transpose_left_4: fixes u :: "'a"   and v :: "'a"   and \<nu> :: "'a" assumes "ide u"   and "ide v"   and "src k = trg v"   and "src k\<^sup>* = trg u"   and "\<guillemotleft>\<nu> : v \<Rightarrow> k\<^sup>* \<star> u\<guillemotright>" shows "k.trnl\<^sub>\<eta> v (k.trnl\<^sub>\<epsilon> u \<nu>) = \<nu>"
heap_is_wellformed_def: shows "heap_is_wellformed \<equiv> l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_heap_is_wellformed get_child_nodes get_disconnected_nodes"
mulex1_mono: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and M :: "'b multiset"   and N :: "'b multiset" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; P x y\<rbrakk> \<Longrightarrow> Q x y"   and "M \<in> multisets A"   and "N \<in> multisets A"   and "mulex1 P M N" shows "mulex1 Q M N"
jumpF_pathstart_eq_0: fixes g :: "real \<Rightarrow> complex"   and z :: "complex" assumes "path g"   and "Re (pathstart g) \<noteq> Re z" shows "jumpF_pathstart g z = 0"
sel_1: fixes x1 :: "'b"   and x2 :: "'c"   and x3 :: "nat"   and x4 :: "('b, 'c) SkewBinomialTree list" shows "val (Node x1 x2 x3 x4) = x1"
t2_in_higher_levels: shows "t2 \<in> Dags (fnct ` Nodes (Suc n) levellista) low high"
Rcd_type1': fixes \<Gamma> :: "binding list"   and fs :: "(char list \<times> trm) list"   and fTs :: "(char list \<times> POPLmarkRecord.type) list" assumes "\<Gamma> \<turnstile> Rcd fs : RcdT fTs" shows "\<forall>(l, U)\<in>set fTs. \<exists>u. fs\<langle>l\<rangle>\<^sub>? = \<lfloor>u\<rfloor> \<and> \<Gamma> \<turnstile> u : U"
disc_9: fixes x3 :: "nat \<times> nat" shows "is_SAT_PUSH' (SAT_PUSH' x3)"
proj2_intersection_unique: fixes l :: "proj2_line"   and m :: "proj2_line"   and p :: "proj2" assumes "l \<noteq> m"   and "proj2_incident p l"   and "proj2_incident p m" shows "p = proj2_intersection l m"
emp_partGen: fixes P :: "'a set set" assumes "{} \<notin> P" shows "{} \<notin> partGen P"
ExecCondFalse: fixes \<sigma> :: "char list \<Rightarrow> nat option"   and e\<^sub>1 :: "expr"   and u\<^sub>1 :: "nat"   and e\<^sub>2 :: "expr"   and u\<^sub>2 :: "nat"   and s\<^sub>2 :: "stmt"   and \<rho> :: "(char list \<Rightarrow> nat option) option"   and s\<^sub>1 :: "stmt" assumes "eval \<sigma> e\<^sub>1 = Some u\<^sub>1"   and "eval \<sigma> e\<^sub>2 = Some u\<^sub>2"   and "u\<^sub>1 \<noteq> u\<^sub>2"   and "s\<^sub>2 \<rhd> \<sigma> \<leadsto> \<rho>" shows "IfEq e\<^sub>1 e\<^sub>2 s\<^sub>1 s\<^sub>2 \<rhd> \<sigma> \<leadsto> \<rho>"
infinite: fixes P :: "nat \<Rightarrow> bool" assumes "eventually P \<U>" shows "infinite {x. P x}"
pseudo_proj_True_same_img: fixes p :: "real"   and M :: "bool stream measure"   and n :: "nat"   and w :: "bool stream"   and x :: "bool stream" assumes "infinite_coin_toss_space p M"   and "pseudo_proj_True n w = pseudo_proj_True n x" shows "stake n w = stake n x"
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"
entailment_implies_validity: fixes S1 :: "'at Literal set set"   and S2 :: "'at Literal set set"   and I :: "'at set" assumes "entails_formula S1 S2"   and "I \<Turnstile> S1" shows "I \<Turnstile> S2"
simps_17: fixes V :: "'a"   and e1 :: "('a, 'a, 'b) exp"   and e2 :: "('a, 'a, 'b) exp" shows "syncvars (sync\<^bsub>V\<^esub> (e1) e2) = (syncvars e1 \<and> syncvars e2 \<and> V \<notin> fv e2)"
msg6_KeyCryptKey_disj: fixes B :: "agent"   and KC3 :: "nat"   and C :: "agent"   and N :: "nat"   and KC2 :: "nat"   and cardSK :: "nat"   and X :: "msg"   and Y :: "msg"   and evs :: "event list" assumes "Gets B \<lbrace>Crypt KC3 \<lbrace>Agent C, Nonce N, Key KC2, Key cardSK, X\<rbrace>, Y\<rbrace> \<in> set evs"   and "cardSK \<notin> symKeys"   and "evs \<in> set_cr" shows "Key cardSK \<in> analz (knows Spy evs) \<or> (\<forall>K. \<not> KeyCryptKey cardSK K evs)"
associatedD2: fixes a :: "'a"   and b :: "'a" assumes "a \<sim> b"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "\<exists>u\<in>Units G. a = b \<otimes> u"
ultra: fixes F :: "'b filter"   and P :: "'b \<Rightarrow> bool" assumes "ultrafilter F" shows "eventually P F \<or> (\<forall>\<^sub>F x in F. \<not> P x)"
local_flow_pend: shows "local_flow (\<lambda>s. fball (- s $ 1) s) UNIV UNIV pend_flow"
valid_cfg_cont: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg"   and s :: "'s \<times> ('c \<Rightarrow> real) set" assumes "cfg \<in> R_G.valid_cfg"   and "s \<in> set_pmf (action cfg)" shows "cont cfg s \<in> R_G.valid_cfg"
q3_def: shows "q3 \<equiv> q1 - monom_mult (lookup q1 (t2 \<oplus> lt f2) / lc f2) t2 f2"
strict_implies_order: fixes b :: "'a list"   and a :: "'a list" assumes "strict_prefix b a" shows "Sublist.prefix b a"
up3_def: fixes a :: "'a"   and sib :: "'a tree"   and twist :: "bool"   and u :: "'a up2" shows "up3 a sib twist u = Time_Monad.val (up3_tm a sib twist u)"
select_convs_1: fixes access :: "acc_modi"   and static :: "bool"   and more :: "'b" shows "static \<lparr>access = access, static = static, \<dots> = more\<rparr> = static"
map_poly_def: fixes f :: "'c \<Rightarrow> 'b"   and p :: "'c poly" shows "map_poly f p = Poly (map f (coeffs p))"
3: shows "\<turnstile> concl (Thm (Imp (Imp p (Imp p p)) (Imp p p)))"
conjugate_conjl1: fixes y :: "'a" shows "conjugate (\<lambda>x. x \<lhd> y) = (\<lambda>x. x \<cdot> y)"
card_chamber: fixes S :: "'b set"   and x :: "'b set set" assumes "CoxeterComplex S"   and "SimplicialComplex.maxsimp (CoxeterComplex.TheComplex S) x" shows "card x = card S"
exhaust: fixes y :: "('a, 'b) rbt" obtains "y = Leaf"   | x21 :: "('a, 'b) rbt" and x22 :: "color" and x23 :: "'a" and x24 :: "'b" and x25 :: "('a, 'b) rbt" where "y = Node x21 x22 x23 x24 x25"
sublist_Nil_right: fixes xs :: "'a list" shows "sublist xs [] = (xs = [])"
simps: fixes xs :: "'b list"   and ys :: "'c list"   and w :: "'c"   and ws :: "'c list"   and z :: "'c"   and zs :: "'c list" shows "proj_list_5 xs (ys, w, ws, z, zs) = (take (length ys) xs, xs ! length ys, take (length ws) (drop (length ys + 1) xs), xs ! (length ys + 1 + length ws), drop (length ys + 1 + length ws + 1) xs)"
modulo_mod_ring_def: fixes x :: "'b mod_ring"   and y :: "'b mod_ring" shows "x mod y = (if y = 0 then x else 0)"
pow_one: fixes b :: "'b"   and x :: "nat" assumes "(0::'b) < b"   and "b < (1::'b)" shows "b ^ x \<le> (1::'b)"
oprod_Preorder: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set" assumes "Preorder r"   and "Preorder r'"   and "antisym r"   and "antisym r'" shows "Preorder (r *o r')"
path_cons_simp': fixes v :: "'d"   and v' :: "'d"   and e :: "('d \<times> 'c) \<times> 'd \<times> 'b"   and pth :: "(('d \<times> 'c) \<times> 'd \<times> 'b) list" shows "path v v' (e # pth) = (\<exists>b a ba. e = ((v, b), a, ba) \<and> ((v, b), a, ba) \<in> edges \<and> path a v' pth)"
GSMP_is_homogeneous: fixes \<A> :: "('b strand_label \<times> ('fun, 'var) strand_step) list"   and Sec :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term" assumes "\<forall>l l'. l \<noteq> l' \<longrightarrow> GSMP_disjoint (trms\<^sub>l\<^sub>s\<^sub>t (proj l \<A>)) (trms\<^sub>l\<^sub>s\<^sub>t (proj l' \<A>)) Sec"   and "t \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t \<A>)"   and "t \<notin> Sec" shows "homogeneous\<^sub>l\<^sub>s\<^sub>t t \<A> Sec"
fst_image_Sigma: fixes A :: "'a set"   and B :: "'a \<Rightarrow> 'b set" shows "fst ` Sigma A B = {x \<in> A. B x \<noteq> {}}"
akra_bazzi_function_def: fixes x\<^sub>0 :: "nat"   and x\<^sub>1 :: "nat"   and k :: "nat"   and as :: "real list"   and bs :: "real list"   and ts :: "(nat \<Rightarrow> nat) list"   and f :: "nat \<Rightarrow> real"   and integrable :: "(real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> real \<Rightarrow> bool"   and integral :: "(real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> real \<Rightarrow> real"   and g :: "nat \<Rightarrow> real" shows "akra_bazzi_function x\<^sub>0 x\<^sub>1 k as bs ts f integrable integral g \<equiv> akra_bazzi_recursion x\<^sub>0 x\<^sub>1 k as bs ts \<and> akra_bazzi_function_axioms x\<^sub>0 x\<^sub>1 k as ts f integrable integral g"
learn_bc_prenum_eventually_zero: fixes b :: "nat" shows "\<exists>k. \<phi> (the (s (e_append_zeros b k))) (e_length b + k) \<down>= 0"
\<epsilon>_def: shows "Adj.\<epsilon> \<equiv> transformation_by_components.map (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) Adj.FG.map Adj.\<epsilon>o"
prefix_eq_prefix_take: fixes xs :: "'d list"   and ys :: "'d list" shows "prefix xs ys = (ys \<down> length xs = xs)"
affine_hull_closed_segment: fixes a :: "'a"   and b :: "'a" shows "affine hull closed_segment a b = affine hull {a, b}"
finite_polytope_faces: fixes S :: "'b set" assumes "polytope S" shows "finite {F. F face_of S}"
that: fixes h' :: "heap"   and r :: "'b" assumes "effect f h h' r"   and "effect (g r) h' h'' r'" shows "thesis"
assms_6: shows "\<forall>y\<in>A. \<forall>x\<in>A. gt y x \<longrightarrow> gt (f y) (f x)"
chines_eq_2: shows "HH\<mu>H\<nu>\<pi>\<rho>.chine = assoc\<mu>\<nu>\<pi>.cod.leg0 \<down>\<down> \<rho>.leg1"
of_int_less_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and w :: "int"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (ring_1.of_int one plus zero minus w) (ring_1.of_int one plus zero minus z) = (w < z)"
float_def: shows "float = {real_of_int m * 2 powr real_of_int e |m e. True}"
finite_fun_graph: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "finite A" shows "finite (fun_graph f \<inter> A \<times> UNIV)"
convex_explicit: fixes S :: "'a set" shows "convex S = (\<forall>t u. finite t \<and> t \<subseteq> S \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> sum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *\<^sub>R x) \<in> S)"
approximating_fun_imp_semantics: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and p :: "'d"   and rs :: "'c rule list"   and s :: "state"   and t :: "state" assumes "wf_ruleset \<gamma> p rs"   and "approximating_bigstep_fun \<gamma> p rs s = t" shows "\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow>\<^sub>\<alpha> t"
inverse_arrows: fixes \<mu> :: "'a arrow_of_spans_data" assumes "arr \<mu>"   and "C.iso (Chn \<mu>)" shows "inverse_arrows \<mu> \<lparr>Chn = C.inv (Chn \<mu>), Dom = Cod \<mu>, Cod = Dom \<mu>\<rparr>"
hyp2_axiom3: shows "\<forall>a b c. a b \<congruent>\<^sub>K c c \<longrightarrow> a = b"
bind_bind_conv_pair_spmf: fixes p :: "'b spmf"   and q :: "'c spmf"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a spmf" shows "p \<bind> (\<lambda>x. q \<bind> f x) = pair_spmf p q \<bind> (\<lambda>(x, y). f x y)"
negint: shows "integrable M (expl_cond_expect M Y (\<lambda>w. max 0 (- X w)))"
eAbs_simp1: fixes xs :: "'varSort"   and s :: "'sort"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and X :: "'c"   and x :: "'var" assumes "isInBar (xs, s)"   and "igWls MOD s X" shows "eAbs MOD xs x (OK X) = OK (igAbs MOD xs x X)"
assms_5: shows "recfn (Suc n) f\<^sub>2"
mult_isol_equiv_subdistl: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.near_dioid plus times less_eq less" shows "(\<forall>x y z. less_eq x y \<longrightarrow> less_eq (times z x) (times z y)) = (\<forall>x y z. less_eq (times z x) (times z (plus x y)))"
clop_extensive: fixes f :: "'b \<Rightarrow> 'b" assumes "clop f" shows "id \<le> f"
induct: fixes x1 :: "'b"   and x2 :: "'b"   and succs :: "'b \<Rightarrow> 'b list"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "x1 [succs]\<rightarrow>* x2"   and "\<And>g. P g g"   and "\<And>g g' g''. \<lbrakk>g' \<in> set (succs g); g' [succs]\<rightarrow>* g''; P g' g''\<rbrakk> \<Longrightarrow> P g g''" shows "P x1 x2"
fv_subset: fixes t :: "('c, 'd) Term.term"   and M :: "('c, 'd) Term.term set" assumes "t \<in> M" shows "fv t \<subseteq> fv\<^sub>s\<^sub>e\<^sub>t M"
enum_works_finite: fixes N :: "nat set" assumes "finite N" shows "N = enum N ` {..<card N} \<and> strict_mono_on (enum N) {..<card N}"
last_in_set: fixes as :: "'b list" assumes "as \<noteq> \<epsilon>" shows "last as \<in> set as"
simps_4: fixes a :: "name"   and x :: "'d"   and b :: "name" assumes "a \<sharp> x"   and "b \<sharp> x" shows "[(a, b)] \<bullet> x = x"
npos_len: fixes u :: "'b list" assumes "\<^bold>|u\<^bold>| \<le> 0" shows "u = \<epsilon>"
lsc_at_open_mem: fixes x0 :: "'b"   and f :: "'b \<Rightarrow> 'c"   and S :: "'c set" assumes "lsc_at x0 f"   and "open S \<and> f x0 \<in> S" obtains T :: "'b set" where "open T \<and> x0 \<in> T \<and> (\<forall>x'\<in>T. f x' \<le> f x0 \<longrightarrow> f x' \<in> S)"
card_eq_UNIV_imp_eq_UNIV: fixes A :: "'b set" assumes "finite UNIV"   and "card A = CARD('b)" shows "A = UNIV"
code: fixes b :: "'b"   and xs :: "'a llist" shows "tllist_of_llist b xs = (case xs of LNil \<Rightarrow> TNil b | LCons x xs' \<Rightarrow> TCons x (tllist_of_llist b xs'))"
ereal_tendsto_simps_5: fixes f :: "'a \<Rightarrow> real"   and a :: "real"   and F :: "'a filter" shows "((ereal \<circ> f) \<longlongrightarrow> ereal a) F = (f \<longlongrightarrow> a) F"
classes_above_subset2: fixes C :: "char list"   and P :: "'b prog"   and P' :: "'b prog" assumes "classes_above P C \<inter> classes_changed P P' = {}" shows "classes_above P' C \<subseteq> classes_above P C"
falsifies\<^sub>g_empty: fixes C :: "fterm literal set" assumes "falsifies\<^sub>g [] C" shows "C = {}"
bs_cons: fixes w :: "bval"   and v :: "exp"   and \<rho> :: "(nat \<times> bval) list"   and \<rho>' :: "(nat \<times> exp) list"   and x :: "nat" assumes "bs_val w v"   and "bs_env \<rho> \<rho>'" shows "bs_env ((x, w) # \<rho>) ((x, v) # \<rho>')"
set_stake_snth: fixes x :: "'b"   and n :: "nat"   and xs :: "'b stream" shows "(x \<in> set (stake n xs)) = (\<exists>i<n. xs !! i = x)"
apath_awalk_to_apath: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c" assumes "wf_digraph G"   and "pre_digraph.awalk G u p v" shows "pre_digraph.apath G u (wf_digraph.awalk_to_apath G p) v"
aprimedivisor_primepow_2: fixes p :: "'a"   and n :: "'a" assumes "prime p"   and "p dvd n"   and "primepow n" shows "aprimedivisor n = p"
unitary11_gen_cis_blaschke: fixes k :: "complex"   and M :: "complex \<times> complex \<times> complex \<times> complex"   and a :: "complex"   and b :: "complex" assumes "k \<noteq> 0"   and "M = k *\<^sub>s\<^sub>m (a, b, cnj b, cnj a)"   and "a \<noteq> 0"   and "mat_det (a, b, cnj b, cnj a) \<noteq> 0" shows "\<exists>k' \<phi> a'. k' \<noteq> 0 \<and> a' * cnj a' \<noteq> 1 \<and> M = k' *\<^sub>s\<^sub>m (cis \<phi>, 0, 0, 1) *\<^sub>m\<^sub>m (1, - a', - cnj a', 1)"