hcju/isabelleps huggingface dataset:
fold_map_idx_count_2: fixes f :: "nat \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> bool \<times> 'a"   and g :: "nat \<Rightarrow> 'a \<Rightarrow> nat"   and y :: "nat"   and i :: "nat"   and c :: "bool"   and xs :: "'a list" assumes "\<And>i c x y. fst (f i c x) = (c \<or> g y (snd (f i c x)) < g y x)"   and "\<And>i c x y. g y (snd (f i c x)) \<le> g y x"   and "\<And>x y y'. y < y' \<Longrightarrow> g y x < g y' x" shows "foldl g y (snd (fold_map_idx f i c xs)) \<le> foldl g y xs"
emb_step_arg: fixes t :: "('a, 'b) tm" assumes "is_App t" shows "t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b arg t"
0: shows "Cring_Poly.compose R f g = to_polynomial R a \<otimes>\<^bsub>P\<^esub> g [^]\<^bsub>P\<^esub> n"
inverts_components: fixes a :: "'a" assumes "ide a" shows "inverse_arrows (\<rho> a) (\<rho>' a)"
check_cs_ok: fixes cs :: "('a \<times> constraint) list"   and J :: "'a set"   and s :: "QDelta ns_constraint list \<times> (('a, ('a \<times> QDelta atom) list) mapping \<times> ((nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping) \<times> 'a list) \<times> ('a, QDelta) state"   and s' :: "QDelta ns_constraint list \<times> (('a, ('a \<times> QDelta atom) list) mapping \<times> ((nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping) \<times> 'a list) \<times> ('a, QDelta) state" assumes "invariant_simplex' cs J s"   and "check_simplex' s = Inr s'" shows "checked_simplex' cs J s'"
odd_Suc_minus_one: fixes n :: "nat" assumes "odd n" shows "Suc (n - Suc 0) = n"
vsubstEnv_updEnv_fresh: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and y :: "'var"   and y1 :: "'var"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and x :: "'var"   and xs :: "'varSort" assumes "good X"   and "fresh ys y X" shows "rho [x \<leftarrow> X]_xs &[y1 // y]_ys = rho &[y1 // y]_ys [x \<leftarrow> X]_xs"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" assumes "l_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs"   and "l_set_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_tag_name set_tag_name_locs" shows "l_set_tag_name_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs set_tag_name set_tag_name_locs"
beta_preserves_term_ok': fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and r :: "Core.term"   and s :: "Core.term" assumes "term_ok' \<Sigma> r"   and "r \<rightarrow>\<^sub>\<beta> s" shows "term_ok' \<Sigma> s"
one_eq: shows "\<one>\<^bsub>R\<^esub> = 1"
is_path_memb: fixes G :: "('a, 'b) graph"   and v :: "'a"   and p :: "('a \<times> 'b \<times> 'a) list"   and v' :: "'a" assumes "valid_graph G"   and "valid_graph.is_path G v p v'" shows "v \<in> nodes G \<and> v' \<in> nodes G"
lossless_R1_xor: fixes A :: "bool \<times> bool"   and B :: "bool \<times> bool" shows "lossless_spmf (R1_xor A B)"
node_deliverTE: fixes i :: "nat"   and s :: "'c"   and R :: "nat set"   and d :: "nat"   and s' :: "'c"   and S :: "('c \<times> 'd seq_action \<times> 'c) set" assumes "(NodeS i s R, i:deliver(d), NodeS i s' R) \<in> node_sos S" obtains "(s, deliver d, s') \<in> S"
\<theta>\<^sub>h_def: shows "\<theta>\<^sub>h = tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0"
zero_order_3: fixes n :: "'c multiset" shows "\<not> n \<subset># {#}"
PHILs\<^sub>n\<^sub>o\<^sub>r\<^sub>m_rec: fixes n :: "nat" shows "PHILs\<^sub>n\<^sub>o\<^sub>r\<^sub>m n = (\<lambda>ps. Mprefix (Tr\<^sub>P n ps) (\<lambda>e. PHILs\<^sub>n\<^sub>o\<^sub>r\<^sub>m n (Up\<^sub>P ps e)))"
inputChainInject: fixes xvec :: "name list"   and M :: "'a"   and P :: "('a, 'b, 'c) psi"   and N :: "'a"   and Q :: "('a, 'b, 'c) psi" shows "(inputChain xvec M P = inputChain xvec N Q) = (M = N \<and> P = Q)"
assms_3: fixes Y :: "'a set" assumes "Complete_Partial_Order.chain (\<le>) Y"   and "Y \<noteq> {}"   and "\<And>x. x \<in> Y \<Longrightarrow> \<not> x \<le> bound" shows "f (\<Squnion> Y) = \<Or> (f ` Y)"
assms_2: shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, t\<rangle> \<Rightarrow> t'"
upto_enum_word: fixes x :: "'a word"   and y :: "'a word" shows "[x .e. y] = map word_of_nat [unat x..<Suc (unat y)]"
f_chain: shows "\<forall>i\<in>{m..<n}. (f (Suc i), f i) \<in> R"
prems_3: fixes i :: "'a" assumes "i \<in> set (a # xs)" shows "i < e"
Abs: fixes T :: "POPLmark.type"   and t :: "trm" shows "(\<lambda>:T. t) \<in> value"
cs_return: fixes \<pi> :: "nat \<Rightarrow> 'n"   and n :: "nat" assumes "\<pi> n = local.return" shows "cs\<^bsup>\<pi>\<^esup> n = [\<pi> n]"
sorted_group_sort: fixes xs :: "(primfun \<times> real) list" shows "sorted (map fst (groupsort_primfun.group_sort xs))"
step: fixes x :: "'a"   and a :: "'a" assumes "x \<in> iterates_above a" shows "f x \<in> iterates_above a"
RECT_unfold: fixes B :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> 'd" assumes "trimono B" shows "REC\<^sub>T B = B (REC\<^sub>T B)"
composite_in_hom_1: shows "\<guillemotleft>cod \<mu> \<star> cod \<nu> \<star> cod \<pi> : local.dom (cod \<mu>) \<star> local.dom (cod \<nu>) \<star> local.dom (cod \<pi>) \<Rightarrow> cod (cod \<mu>) \<star> cod (cod \<nu>) \<star> cod (cod \<pi>)\<guillemotright>"
inorder_delete: fixes t :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and x :: "'c" assumes "sorted1 (Tree2.inorder t)" shows "Tree2.inorder (delete x t) = AList_Upd_Del.del_list x (Tree2.inorder t)"
assms_1: shows "reconstruction sl_impl m2 state u (Polynomial.smult lu u) lu d r vs res cands = fs"
assms: shows "stfx.rel (V, q) (V', q')"
poly_roots_set_same: fixes a :: "real"   and b :: "real"   and c :: "real" shows "{x. a * x\<^sup>2 + b * x + c = 0} = {x. poly [:c, b, a:] x = 0}"
array_map_def: fixes f :: "nat \<Rightarrow> 'b \<Rightarrow> 'a"   and a :: "'b array" shows "array_map f a = array_of_list (map (\<lambda>(i, v). f i v) (assoc_list_of_array a))"
fps_numeral_mult_divide: fixes b :: "num"   and x :: "'b fps"   and c :: "num" shows "numeral b * x / numeral c = numeral b / numeral c * x"
subset_entailed: fixes N2 :: "'f set"   and N1 :: "'f set" assumes "N2 \<subseteq> N1" shows "N1 \<Turnstile> N2"
map_pmf_def: fixes f :: "'d \<Rightarrow> 'c"   and M :: "'d pmf" shows "map_pmf f M = M \<bind> (\<lambda>x. return_pmf (f x))"
keys_simps_2: fixes c :: "color"   and l :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b"   and r :: "('a, 'b) rbt" shows "keys (Branch c l k v r) = keys l @ k # keys r"
crel_vs_iterate_state: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and g :: "'d \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> ('b, 'f) state"   and xs :: "'d list" assumes "((=) ===>\<^sub>T R) g f" shows "crel_vs (=) () (iterate_state f xs)"
right_idem: fixes a :: "'c"   and b :: "'c" shows "a + b + b = a + b"
invimage_def: fixes D :: "V" shows "invimage D = VLambda D (vsplit (\<lambda>r. (`\<^sub>\<circ>) (r\<inverse>\<^sub>\<circ>)))"
get_scdom_component_ptrs_same_scope_component: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and sc :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "DocumentClass.type_wf h"   and "DocumentClass.known_ptrs h"   and "h \<turnstile> get_scdom_component ptr \<rightarrow>\<^sub>r sc"   and "ptr' \<in> set sc" shows "h \<turnstile> get_scdom_component ptr' \<rightarrow>\<^sub>r sc"
rank_of_sub_cong: fixes \<E> :: "'a set"   and \<E>' :: "'a set" assumes "\<E> \<subseteq> \<E>"   and "\<E>' \<subseteq> \<E>" shows "indep_system.lower_rank_of (\<E>.indep_in \<E>) \<E>' = \<E>.rank_of \<E>'"
continuous_on_const: fixes s :: "'e set"   and c :: "'f" shows "continuous_on s (\<lambda>x. c)"
ForAllIteratorTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and src :: "'a expr"   and k :: "call_kind"   and its :: "String.literal list"   and its_ty :: "'a OCL_Types.type option"   and body :: "'a expr"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E ForAllIteratorCall src k its its_ty body : \<tau>" obtains \<tau>' :: "'a OCL_Types.type" and \<sigma> :: "'a OCL_Types.type" where "k = ArrowCall" and "\<Gamma> \<turnstile>\<^sub>I (src, its, its_ty, body) : (\<tau>', \<sigma>, \<tau>)" and "\<tau> \<le> Boolean[?]"
initial_of_def: shows "initial_of \<equiv> id \<circ> Record.iso_tuple_fst Record.tuple_iso_tuple \<circ> Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> Record.iso_tuple_fst strips_problem_ext_Tuple_Iso"
inner_ge_zero: fixes x :: "('a, 'n) vec" shows "0 \<le> real_of (inner x x)"
higher_differentiable_on_imp_differentiable_on: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and k :: "nat" assumes "higher_differentiable_on S f k"   and "0 < k" shows "f differentiable_on S"
simps_2: fixes x2 :: "'h"   and y2 :: "'h" shows "(Inr x2 = Inr y2) = (x2 = y2)"
simps: fixes cs :: "(symbol \<times> character list) list" shows "charslength cs = length (chars cs)"
from_nat_mono': fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Rep :: "'d \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'd"   and i :: "nat"   and j :: "nat" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "i \<le> j"   and "j < CARD('d)" shows "less_eq (mod_type.from_nat Abs i) (mod_type.from_nat Abs j)"
IH_2: fixes x :: "'a" assumes "x \<in> A" shows "random_bst (R x) = map_pmf bst_of_list (pmf_of_set (permutations_of_set (R x)))"
simps_7: fixes t :: "'a local.term" shows "Dom \<^bold>\<r>\<^bold>[t\<^bold>] = Dom t \<^bold>\<star> Src t"
folding': fixes C :: "'a set" assumes "C \<in> f \<turnstile> \<C>" shows "\<exists>!D. D \<in> \<C> - f \<turnstile> \<C> \<and> f ` D = C"
simps: fixes e1 :: "'a"   and a1 :: "'b"   and r1 :: "nat"   and ts1 :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and e2 :: "'a"   and a2 :: "'b"   and r2 :: "nat"   and ts2 :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "SkewBinomialHeapStruc.link (SkewBinomialHeapStruc.Node e1 a1 r1 ts1) (SkewBinomialHeapStruc.Node e2 a2 r2 ts2) = (if a1 \<le> a2 then SkewBinomialHeapStruc.Node e1 a1 (Suc r1) (SkewBinomialHeapStruc.Node e2 a2 r2 ts2 # ts1) else SkewBinomialHeapStruc.Node e2 a2 (Suc r2) (SkewBinomialHeapStruc.Node e1 a1 r1 ts1 # ts2))"
exhaust: fixes y :: "'b local.term" obtains x1 :: "'b" where "y = \<^bold>\<langle>x1\<^bold>\<rangle>"   | "y = \<^bold>\<I>"   | x31 :: "'b local.term" and x32 :: "'b local.term" where "y = x31 \<^bold>\<otimes> x32"   | x41 :: "'b local.term" and x42 :: "'b local.term" where "y = x41 \<^bold>\<cdot> x42"   | x5 :: "'b local.term" where "y = \<^bold>\<l>\<^bold>[x5\<^bold>]"   | x6 :: "'b local.term" where "y = \<^bold>\<l>\<^sup>-\<^sup>1\<^bold>[x6\<^bold>]"   | x7 :: "'b local.term" where "y = \<^bold>\<r>\<^bold>[x7\<^bold>]"   | x8 :: "'b local.term" where "y = \<^bold>\<r>\<^sup>-\<^sup>1\<^bold>[x8\<^bold>]"   | x91 :: "'b local.term" and x92 :: "'b local.term" and x93 :: "'b local.term" where "y = \<^bold>\<a>\<^bold>[x91, x92, x93\<^bold>]"   | x101 :: "'b local.term" and x102 :: "'b local.term" and x103 :: "'b local.term" where "y = \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[x101, x102, x103\<^bold>]"
inj_image_mem_iff: fixes f :: "'b \<Rightarrow> 'c"   and a :: "'b"   and A :: "'b set" assumes "inj f" shows "(f a \<in> f ` A) = (a \<in> A)"
simps_2: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and ys :: "'b list" shows "sorted_wrt P (x # ys) = (Ball (set ys) (P x) \<and> sorted_wrt P ys)"
node_lift_anycast_statelessassm: fixes T :: "((nat \<Rightarrow> 'a) \<times> 'b, 'c seq_action) automaton"   and I :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> bool"   and i :: "nat"   and Q :: "(nat \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> bool"   and R\<^sub>i :: "nat set" assumes "T \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. orecvmsg I \<sigma>, other (\<lambda>_ _. True) {i} \<rightarrow>) globala (\<lambda>(\<sigma>, a, \<sigma>'). anycast (Q \<sigma> \<sigma>') a)" shows "\<langle>i : T : R\<^sub>i\<rangle>\<^sub>o \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg I \<sigma>, other (\<lambda>_ _. True) {i} \<rightarrow>) globala (\<lambda>(\<sigma>, a, \<sigma>'). castmsg (Q \<sigma> \<sigma>') a)"
kcomp_idl: fixes f :: "'a \<Rightarrow> 'b set" shows "\<eta> \<circ>\<^sub>K f = f"
span_clauses_4: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'d"   and S :: "'d set"   and c :: "'c" assumes "module scale"   and "x \<in> module.span scale S" shows "scale c x \<in> module.span scale S"
heap_WHILET_unfold: fixes b :: "'b \<Rightarrow> bool Heap"   and f :: "'b \<Rightarrow> 'b Heap"   and s :: "'b" shows "heap_WHILET b f s = b s \<bind> (\<lambda>bv. if bv then f s \<bind> heap_WHILET b f else return s)"
simps_5: fixes n :: "nat"   and execs :: "'dom_t \<Rightarrow> 'action_t list list"   and s :: "'state_t" assumes "\<not> interrupt (Suc n)"   and "\<not> thread_empty (execs (current s))"   and "\<not> precondition (next_state s execs) (next_action s execs)" shows "run (Suc n) (Some s) execs = None"
fwi_cyc_free_diag: fixes n :: "nat"   and I :: "nat set"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat"   and i :: "nat" assumes "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> I \<longrightarrow> (0::'a) \<le> len m i i xs"   and "(0::'a) \<le> m k k"   and "k \<le> n"   and "k \<in> I"   and "i \<le> n" shows "(0::'a) \<le> fwi m n k n n i i"
gen_dfs_fe_inv_imp_post: fixes E :: "('a \<times> 'a) set"   and U :: "'a set"   and S :: "'a set"   and V :: "'a set"   and u0 :: "'a"   and V' :: "'a set"   and cyc :: "bool"   and cyc' :: "bool" assumes "gen_dfs_pre E U S V u0"   and "gen_dfs_fe_inv E U (insert u0 S) (insert u0 V) u0 {} V' cyc"   and "cyc \<longrightarrow> cyc'" shows "gen_dfs_post E U S V u0 V' cyc'"
se_star_append: fixes c1 :: "('c, 'd) conf"   and ls1 :: "('c, 'd) label list"   and ls2 :: "('c, 'd) label list"   and c2 :: "('c, 'd) conf" shows "se_star c1 (ls1 @ ls2) c2 = (\<exists>c. se_star c1 ls1 c \<and> se_star c ls2 c2)"
axioms_2: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e" assumes "natural_transformation A B F G \<tau>" shows "category B"
arr_char: fixes f :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr" shows "S.VV.arr f = (S.arr (fst f) \<and> S.arr (snd f) \<and> S.src (fst f) = S.trg (snd f))"
assms_2: fixes i :: "nat" shows "f i \<in> A"
distinct_17: fixes x3 :: "nat"   and x51 :: "Core.term"   and x52 :: "Core.term" shows "Bv x3 \<noteq> x51 $ x52"
simps_2: fixes x :: "'a"   and ys :: "'a list"   and z :: "'a"   and zs :: "'a list" shows "replace x ys (z # zs) = (if z = x then ys @ zs else z # replace x ys zs)"
abs_eq: fixes x :: "'b" shows "vector_to_cblinfun x = CBlinfun (\<lambda>\<phi>. one_dim_iso \<phi> *\<^sub>C x)"
obj_is_self_adjoint_1: fixes a :: "'a" assumes "obj a" shows "adjoint_pair a a"
semiconfluentp_cancel1: shows "semiconfluentp cancel1\<inverse>\<inverse>"
i_7: shows "h \<in> insert x F \<rightarrow> carrier G"
fpair_iff: fixes a :: "V"   and b :: "V"   and a' :: "V"   and b' :: "V" shows "([a, b]\<^sub>\<circ> = [a', b']\<^sub>\<circ>) = (a = a' \<and> b = b')"
cases_scheme: fixes r :: "('c, 'd) monoid_scheme" obtains carrier :: "'c set" and mult :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" and one :: "'c" and more :: "'d" where "r = \<lparr>carrier = carrier, monoid.mult = mult, one = one, \<dots> = more\<rparr>"
T_snoc: fixes rs :: "'c list"   and as :: "(nat \<times> nat list) list"   and init :: "'c list"   and r :: "'c"   and a :: "nat \<times> nat list" assumes "length rs = length as" shows "T\<^sub>p init (rs @ [r]) (as @ [a]) = T\<^sub>p init rs as + t\<^sub>p (steps' init rs as (length rs)) r a"
Polygamma_approx_Suc: fixes j :: "nat"   and m :: "nat" shows "Polygamma_approx (Suc j) m = deriv (Polygamma_approx j m)"
monomial_function_def: fixes R :: "('c, 'd) monoid_scheme"   and c :: "'c"   and n :: "nat" shows "monomial_function R c n = (\<lambda>x\<in>carrier R. c \<otimes>\<^bsub>R\<^esub> x [^]\<^bsub>R\<^esub> n)"
all_ideal_is_principal: fixes I :: "'a set" assumes "ideal I" shows "principal_ideal I"
expansion_factor_linear: fixes p :: "proj2"   and q :: "proj2"   and r :: "proj2"   and J :: "cltn2"   and k :: "real" assumes "p \<in> hyp2 \<union> S"   and "q \<in> hyp2 \<union> S"   and "r \<in> hyp2 \<union> S"   and "is_K2_isometry J"   and "cart2_pt r = k *\<^sub>R cart2_pt p + (1 - k) *\<^sub>R cart2_pt q" shows "expansion_factor r J = k * expansion_factor p J + (1 - k) * expansion_factor q J"
exists_index: fixes t :: "('a, 'b, 'c) event list"   and ev :: "('a, 'b, 'c) event"   and j :: "nat"   and i :: "nat" assumes "trace init t final"   and "ev \<in> set (take (j - i) (drop i t))" shows "\<exists>k\<ge>i. k < j \<and> ev = t ! k"
tendsto_nonzero_has_sgnx: fixes f :: "real \<Rightarrow> real"   and c :: "real"   and F :: "real filter" assumes "(f \<longlongrightarrow> c) F"   and "c \<noteq> 0" shows "(f has_sgnx sgn c) F"
left_adjoint_extends_to_adjoint_pair: fixes f :: "'a" assumes "is_left_adjoint f" shows "adjoint_pair f f\<^sup>*"
alphaAbs_trans: fixes A :: "('e, 'f, 'varSort, 'var, 'g) qAbs"   and A' :: "('e, 'f, 'varSort, 'var, 'g) qAbs"   and A'' :: "('e, 'f, 'varSort, 'var, 'g) qAbs" assumes "qGoodAbs A \<or> qGoodAbs A' \<or> qGoodAbs A''"   and "A $= A'"   and "A' $= A''" shows "A $= A''"
Match_def: fixes x3 :: "name"   and x2 :: "name"   and x1 :: "pi" shows "[x3\<frown>x2]x1 \<equiv> Abs_pi (Match_Rep x3 x2 (Rep_pi x1))"
module_hom_def: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and f :: "'e \<Rightarrow> 'f" shows "module_hom s1 s2 f \<equiv> module s1 \<and> module s2 \<and> module_hom_axioms s1 s2 f"
has_density_dens_AE: fixes f :: "'c \<Rightarrow> ennreal"   and f' :: "'c \<Rightarrow> ennreal"   and N :: "'c measure"   and M :: "'c measure" assumes "AE y in N. f y = f' y"   and "f' \<in> borel_measurable N"   and "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f' x"   and "has_density M N f" shows "has_density M N f'"
simps_4: fixes \<theta> :: "(name \<times> trm) list"   and t :: "trm" shows "\<theta><[t]> = [\<theta><t>]"
antilinear_imp_scaleC: fixes D :: "complex \<Rightarrow> 'd" assumes "antilinear D" obtains d :: "'d" where "D = (\<lambda>x. cnj x *\<^sub>C d)"
principal_not_totally_separable: assumes "principal_dchar n = principal_dchar n" shows "\<not> (\<forall>k>0. principal.separable k)"
unique_limit: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"   and "a \<in> carrier Zp"   and "b \<in> carrier Zp"   and "Zp_converges_to s a"   and "Zp_converges_to s b" shows "a = b"
assms_4: fixes \<sigma> :: "'a"   and p :: "'b"   and a :: "'c"   and \<sigma>' :: "'a"   and p' :: "'b" assumes "(\<sigma>, p) \<in> oreachable A S U"   and "P \<sigma> p"   and "((\<sigma>, p), a, \<sigma>', p') \<in> automaton.trans A"   and "S \<sigma> \<sigma>' a" shows "P \<sigma>' p'"
set_mono_suffix: fixes xs :: "'c list"   and ys :: "'c list" assumes "suffix xs ys" shows "set xs \<subseteq> set ys"
zip_Cons1: fixes x :: "'c"   and xs :: "'c list"   and ys :: "'d list" shows "zip (x # xs) ys = (case ys of [] \<Rightarrow> [] | y # ys \<Rightarrow> (x, y) # zip xs ys)"
Range_Un_eq: fixes A :: "('d \<times> 'c) set"   and B :: "('d \<times> 'c) set" shows "Range (A \<union> B) = Range A \<union> Range B"
ctr: fixes x :: "quat" shows "- x = Quat (- quat.Re x) (- Im1 x) (- Im2 x) (- Im3 x)"
compute_\<ff>s_def: fixes s :: "'a list" shows "compute_\<ff>s s = (let \<ff>s = replicate (length s + 1) 0; i = 0; j = 1 in WHILE\<^sub>T\<^bsup>I_out_cb s\<^esup> (\<lambda>(\<ff>s, uu_, j). j < length \<ff>s) (\<lambda>(\<ff>s, i, j). WHILE\<^sub>T\<^bsup>I_in_cb s j\<^esup> (\<lambda>i. 0 < i \<and> s ! (i - 1) \<noteq> s ! (j - 1)) (\<lambda>i. ASSERT (i - 1 < length \<ff>s) \<bind> (\<lambda>_. Let (\<ff>s ! (i - 1)) RETURN)) i \<bind> (\<lambda>i. let i = i + 1 in ASSERT (j < length \<ff>s) \<bind> (\<lambda>_. let \<ff>s = \<ff>s[j := i]; j = j + 1 in RETURN (\<ff>s, i, j)))) (\<ff>s, i, j) \<bind> (\<lambda>(\<ff>s, uu_, uu_). RETURN \<ff>s))"
vUniform_def: fixes sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and start :: "'a \<Rightarrow> 'b"   and v :: "bool"   and c :: "('a, 'c, 'b) configuration" assumes "flpSystem sends" shows "flpSystem.vUniform trans sends start v c \<equiv> asynchronousSystem.initReachable trans sends start c \<and> (\<forall>p. {v. flpSystem.pSilDecVal trans sends start v p c} = {v})"
assms: shows "p \<in> pmdl (set (init_syzygy_list bs))"
hyps: assumes "i < min (dim_row A) (dim_col A)" shows "A $$ (0, 0) dvd A $$ (i, i)"
fv_in_subst_img: fixes s :: "'d \<Rightarrow> ('c, 'd) Term.term"   and v :: "'d"   and t :: "('c, 'd) Term.term" assumes "s v = t"   and "t \<noteq> Var v" shows "fv t \<subseteq> range_vars s"
user_accessible_def: fixes s :: "'b sparc_state"   and pa :: "word36" shows "user_accessible s pa \<equiv> \<exists>va p. virt_to_phys va (mmu s) (mem s) = Some p \<and> mmu_readable (get_acc_flag (snd p)) 10 \<and> fst p = pa"
denominator_pos: fixes t :: "real" shows "0 < f (2 - t) + f (t - 1)"
comp_left_dist_sup: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and conv :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.bounded_distrib_allegory one times inf less_eq less sup bot top conv" shows "sup (times x y) (times x z) = times x (sup y z)"
update_convs_2: fixes getM' :: "(Literal \<times> bool) list \<Rightarrow> (Literal \<times> bool) list"   and getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and more :: "'a" shows "getM_update getM' \<lparr>getF = getF, getM = getM, \<dots> = more\<rparr> = \<lparr>getF = getF, getM = getM' getM, \<dots> = more\<rparr>"
piecewise_C1_differentiable_D1: fixes g1 :: "real \<Rightarrow> 'b"   and g2 :: "real \<Rightarrow> 'b" assumes "g1 +++ g2 piecewise_C1_differentiable_on {0..1}" shows "g1 piecewise_C1_differentiable_on {0..1}"
array_get_dyn_def: fixes dflt :: "'b"   and a :: "'b array"   and i :: "nat" shows "array_get_dyn dflt a i \<equiv> Array.len a \<bind> (\<lambda>l. if i < l then Array.nth a i else return dflt)"