hcju/isabelleps huggingface dataset:
induct: fixes x1 :: "string_unop"   and x2 :: "'a OCL_Types.type"   and x3 :: "'b OCL_Types.type"   and P :: "string_unop \<Rightarrow> 'a OCL_Types.type \<Rightarrow> 'b OCL_Types.type \<Rightarrow> bool" assumes "string_unop_type x1 x2 x3"   and "P SizeOp String[1] Integer[1]"   and "P CharactersOp String[1] (Sequence String[1])"   and "P ToUpperCaseOp String[1] String[1]"   and "P ToLowerCaseOp String[1] String[1]"   and "P ToBooleanOp String[1] Boolean[1]"   and "P string_unop.ToIntegerOp String[1] Integer[1]"   and "P ToRealOp String[1] Real[1]" shows "P x1 x2 x3"
simps_2: fixes \<sigma> :: "'tp"   and al :: "univ list" shows "TE_intF (Tag \<sigma>) al = hd al"
1: shows "[\<^bold>\<diamond>\<^bold>\<not>p \<^bold>& \<^bold>\<box>q in v]"
simps_3: fixes a :: "'a"   and \<nu> :: "'a set" shows "af_G_letter p(a) \<nu> = (if a \<in> \<nu> then true else false)"
empty_def: shows "BinomialHeap_loc.empty \<equiv> Abs_BinomialHeap []"
that: fixes g :: "complex \<Rightarrow> complex" assumes "g holomorphic_on S"   and "cmod (g a) \<le> pi + cmod (f a)"   and "\<And>z. z \<in> S \<Longrightarrow> f z = cos (g z)" shows "thesis"
c_prod_idl: fixes x :: "'a" shows "1\<^sub>\<pi> \<parallel> x = x"
abs_minus_le_zero: fixes abs :: "'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq (uminus (abs a)) zero"
strict_order_iff: fixes a :: "'c multiset"   and b :: "'c multiset" shows "(a \<subset># b) = (a = a \<inter># b \<and> a \<noteq> b)"
simps_1: shows "Leq_Rel + Leq_Rel = Leq_Rel"
space_prob_algebra: fixes M :: "'b measure" shows "space (prob_algebra M) = {N. sets N = sets M \<and> prob_space N}"
indets_minus_subset: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and q :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "indets (p - q) \<subseteq> indets p \<union> indets q"
THROW_rule_iff: fixes e :: "'c"   and \<Phi> :: "'c \<Rightarrow> bool"   and \<Psi> :: "'d \<Rightarrow> bool" shows "(THROW e \<le> ESPEC \<Phi> \<Psi>) = \<Phi> e"
contracting_def: fixes g :: "nat \<Rightarrow> nat"   and r :: "'b run"   and sub :: "'b run"   and f :: "nat \<Rightarrow> nat" shows "contracting g r sub f \<equiv> contracting_fun g \<and> (\<forall>n. f (g n) \<le> n) \<and> (\<forall>n c k. f (g n) \<le> k \<and> k \<le> n \<longrightarrow> time (Rep_run r k c) = time (Rep_run sub (g n) c)) \<and> (\<forall>n c k. f (g n) < k \<and> k \<le> n \<longrightarrow> \<not> hamlet (Rep_run r k c))"
assms_2: shows "wf\<^sub>t\<^sub>r\<^sub>m t"
finite: fixes n :: "nat"   and m :: "nat" shows "eNorm (\<LL> 2 (lebesgue_on S)) (f n - f m) < \<top>"
fps_nth_cases: fixes y :: "'c list \<Rightarrow> 'd" assumes "y \<in> {f. True}" obtains x :: "('c, 'd) fps" where "y = ($) x"
case_eq_if: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> ('c, 'b) tllist \<Rightarrow> 'a"   and tllist :: "('c, 'b) tllist" shows "(case tllist of TNil x \<Rightarrow> f1 x | TCons x xa \<Rightarrow> f2 x xa) = (if is_TNil tllist then f1 (terminal tllist) else f2 (thd tllist) (ttl tllist))"
ereal_minus_mono: fixes A :: "ereal"   and B :: "ereal"   and D :: "ereal"   and C :: "ereal" assumes "A \<le> B"   and "D \<le> C" shows "A - C \<le> B - D"
prj0_simps_arr: fixes f :: "'a"   and g :: "'a" shows "C.arr \<p>\<^sub>0[f, g] = C.cospan f g"
simps: fixes p :: "int poly"   and l :: "rat"   and r :: "rat" shows "uminus_1 (p, l, r) = (abs_int_poly (poly_uminus p), - r, - l)"
ctxt_def: fixes p :: "prog" shows "ctxt p \<equiv> set (para p)"
rel_fun_trans2': fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and f1 :: "'a \<Rightarrow> 'c"   and f1' :: "'b \<Rightarrow> 'c"   and f2 :: "'a \<Rightarrow> 'd"   and f2' :: "'b \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "rel_fun R (=) f1 f1'"   and "rel_fun R (=) f2 f2'" shows "rel_fun R (=) (\<lambda>x. g (f1 x) (f2 x)) (\<lambda>y. g (f1' y) (f2' y))"
Fvars': fixes \<phi> :: "'fmla"   and x :: "'var" assumes "\<phi> \<in> fmla"   and "x \<in> Fvars \<phi>" shows "x \<in> var"
HFinite_hRe: fixes x :: "complex star" assumes "x \<in> HFinite" shows "hRe x \<in> HFinite"
assms_1: shows "{x, y} \<subseteq> \<V>"
trimmed_const_expansion: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and c :: "real" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term" shows "trimmed (const_expansion c) = (c \<noteq> 0)"
hyps_1: shows "s \<rightarrow>\<^sub>\<beta> t"
extend_stable_project_set: fixes F :: "'a program"   and A :: "'c set" assumes "extend h F \<in> stable A" shows "F \<in> stable (project_set h A)"
shorter: assumes "ys\<^sub>1 = take (length ys\<^sub>1) xs\<^sub>1"   and "ys\<^sub>2 = drop (length ys\<^sub>1) xs\<^sub>1 @ xs\<^sub>2" shows "thesis"
inv_hcomp: fixes \<nu> :: "'b"   and \<mu> :: "'b" assumes "D.iso \<nu>"   and "D.iso \<mu>"   and "src\<^sub>D \<nu> = trg\<^sub>D \<mu>" shows "D.inv (\<nu> \<star>\<^sub>D \<mu>) = D.inv \<nu> \<star>\<^sub>D D.inv \<mu>"
axioms_2: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "regular_weak_composition V H" shows "regular_weak_composition_axioms V H"
diff_numeral_special_11: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and one :: "'c" assumes "class.neg_numeral minus plus zero uminus" shows "minus one (uminus one) = numeral.numeral one plus (num.Bit0 num.One)"
2: fixes \<gamma> :: "'v \<Rightarrow> ('f, 'v) Term.term" assumes "Unifier \<gamma> (s \<cdot> \<theta>) (t \<cdot> \<theta>)" shows "\<delta> \<preceq>\<^sub>\<circ> \<theta> \<circ>\<^sub>s \<gamma>"
conj_upred_def: shows "(\<and>) = (\<squnion>)"
ins2_iff_ins_2: fixes n :: "nat"   and d :: "nat"   and x :: "'a"   and t :: "'a tree"   and t' :: "'a tree" shows "(ins2 n d x t = Bal2 t') = (ins n d x t = Bal t')"
bind_singleton_conv_image: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'c" shows "A \<bind> (\<lambda>x. {f x}) = f ` A"
matching_i_def: fixes i :: "nat"   and V :: "'b set"   and E :: "('b \<times> 'b) set"   and M :: "('b \<times> 'b) set"   and L :: "'b \<Rightarrow> nat" shows "matching_i i V E M L = {e \<in> M. i = 1 \<and> (L (fst e) = i \<or> L (snd e) = i) \<or> 1 < i \<and> L (fst e) = i \<and> L (snd e) = i}"
simps: fixes v1 :: "'v"   and p1 :: "'outPort"   and v2 :: "'v"   and p2 :: "'inPort" shows "edge_begin ((v1, p1), v2, p2) = v1"
preserves_trg: fixes \<mu> :: "'a" assumes "C.arr \<mu>" shows "trg\<^sub>D (F \<mu>) = map\<^sub>0 (trg\<^sub>C \<mu>)"
\<delta>'_def: shows "(\<delta>', p\<^sub>1) = find_closest_bf_code p\<^sub>0 (take 7 (p\<^sub>2 # ps))"
\<epsilon>o_def: fixes x :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" shows "Adj.\<epsilon>o x = \<psi> x (G x)"
\<Phi>_wbalanced: fixes t :: "'a tree" assumes "wbalanced t" shows "\<Phi> t = 0"
list_update_length: fixes xs :: "'b list"   and x :: "'b"   and ys :: "'b list"   and y :: "'b" shows "(xs \<cdot> x # ys)[\<^bold>|xs\<^bold>| := y] = xs \<cdot> y # ys"
fair_chain_impl_consequence_graph: fixes Rs :: "(('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph) set"   and S :: "nat \<Rightarrow> ('d, 'f) labeled_graph" assumes "fair_chain Rs S"   and "\<And>R. R \<in> Rs \<Longrightarrow> subgraph (fst R) (snd R) \<and> finite_graph (fst R)" shows "consequence_graph Rs (chain_sup S)"
connected_def: shows "local.connected \<equiv> \<forall>v\<in>V. \<forall>v'\<in>V. v \<noteq> v' \<longrightarrow> (\<exists>ps. is_path v ps v')"
phiDefs_def: fixes g :: "'a"   and n :: "'node" shows "i.phiDefs g n \<equiv> {v. (n, v) \<in> dom (phis2 g)}"
g_le_f: fixes x :: "'v"   and y :: "'v" assumes "edge \<Delta> x y" shows "g (y, x) \<le> f (x, y)"
inext_eq_infin: fixes a :: "nat"   and I :: "nat set"   and b :: "nat" assumes "inext a I = inext b I"   and "infinite I" shows "a = b"
assms: shows "{i. i < length xs \<and> i \<in> I} = {j}"
map_snd_zip: fixes xs :: "'a list"   and ys :: "'b list" assumes "length xs = length ys" shows "map snd (zip xs ys) = ys"
vsingleton_iff: fixes y :: "V"   and x :: "V" shows "(y \<in>\<^sub>\<circ> ZFC_in_HOL.set {x}) = (y = x)"
simps: fixes l :: "'h"   and A :: "((('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) list"   and B :: "('h strand_label \<times> (('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) stateful_strand_step) list"   and C :: "('h strand_label \<times> (('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) stateful_strand_step) list"   and D :: "('h strand_label \<times> (('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) stateful_strand_step) list"   and E :: "('h strand_label \<times> (('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) stateful_strand_step) list"   and F :: "('h strand_label \<times> (('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) stateful_strand_step) list" shows "transaction_proj l (Transaction A B C D E F) = (let f = proj l in Transaction A (f B) (f C) (f D) (f E) (f F))"
list_allI: fixes l :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "\<And>x. x \<in> set l \<Longrightarrow> P x" shows "list_all P l"
OAlist_tc_lookup_def: fixes xs :: "('b, 'c) oalist_tc" shows "OAlist_tc_lookup xs = lookup_pair_tc (list_of_oalist_tc xs)"
typeof_i32: fixes v :: "v" assumes "typeof v = T_i32" shows "\<exists>c. v = ConstInt32 c"
assms_3: shows "(x \<longlongrightarrow> a) F"
minjec_inj: fixes N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g" assumes "R module N"   and "injec\<^bsub>M,N\<^esub> f" shows "inj_on f (carrier M)"
card_bijections_range_permutation_eq_0: fixes A :: "'c set"   and B :: "'d set" assumes "card A \<noteq> card B" shows "card ({f \<in> A \<rightarrow>\<^sub>E B. bij_betw f A B} // range_permutation A B) = 0"
get_disconnected_document_impl: shows "get_disconnected_document = local.a_get_disconnected_document"
node_deliver_messages_append: fixes xs :: "('msgid \<times> 'op) event list"   and ys :: "('msgid \<times> 'op) event list" shows "node_deliver_messages (xs @ ys) = node_deliver_messages xs @ node_deliver_messages ys"
oexp_eq_0_iff: fixes \<beta> :: "V"   and \<alpha> :: "V" assumes "Ord \<beta>" shows "(\<alpha> \<up> \<beta> = 0) = (\<alpha> = 0 \<and> \<beta> \<noteq> 0)"
induct: fixes P :: "nat \<Rightarrow> bool"   and a0 :: "nat" assumes "\<And>n. n < 200 \<Longrightarrow> P n"   and "\<And>n. \<lbrakk>200 \<le> n; P (nat \<lfloor>real n / 3\<rfloor>); P (nat \<lfloor>3 * real n / 4\<rfloor> + 42)\<rbrakk> \<Longrightarrow> P n" shows "P a0"
rtrancl_into_rtrancl: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "rtranclpR r a b"   and "r b c" shows "rtranclpR r a c"
gAnd_def: fixes v :: "'a gexp"   and va :: "'a gexp" shows "gAnd v va \<equiv> Nor (Nor v v) (Nor va va)"
refl_slice2: fixes n :: "'node SDG_node" assumes "valid_SDG_node n" shows "n \<in> sum_SDG_slice2 n"
target_relation_from_source_target_relation: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set" assumes "\<forall>S T. (SourceTerm S, TargetTerm T) \<in> Rel \<longrightarrow> (TargetTerm (\<lbrakk>S\<rbrakk>), TargetTerm T) \<in> Rel\<^sup>=" shows "\<exists>TRel. (\<forall>T1 T2. (T1, T2) \<in> TRel \<longrightarrow> (TargetTerm T1, TargetTerm T2) \<in> Rel) \<and> (\<forall>T1 T2. (TargetTerm T1, TargetTerm T2) \<in> Rel \<longrightarrow> (T1, T2) \<in> TRel\<^sup>+) \<and> (\<forall>S T. (SourceTerm S, TargetTerm T) \<in> Rel \<longrightarrow> (\<lbrakk>S\<rbrakk>, T) \<in> TRel\<^sup>*)"
prob_space_return: fixes x :: "'c"   and M :: "'c measure" assumes "x \<in> space M" shows "prob_space (return M x)"
\<iota>_is_iso: shows "C.iso \<iota>\<^sub>C"
mcont_sup1: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "mcont Sup less_eq Sup less_eq (sup x)"
nth_append: fixes xs :: "'g list"   and ys :: "'g list"   and n :: "nat" shows "(xs @ ys) ! n = (if n < length xs then xs ! n else ys ! (n - length xs))"
map_def: fixes f :: "'b" shows "\<tau>'\<tau>.map f = (if F.B.arr f then C (\<tau>' (F.B.cod f)) (\<tau> f) else \<tau>.B.null)"
box_def: fixes a :: "'b"   and b :: "'b" shows "box a b = {x. \<forall>i\<in>Basis. a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i}"
lcm_proj1_iff_nat: fixes m :: "nat"   and n :: "nat" shows "(lcm m n = m) = (n dvd m)"
flip_at_base_simps_3: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a \<noteq> c"   and "b \<noteq> c" shows "(a \<leftrightarrow> b) \<bullet> c = c"
simps_1: fixes step :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'b"   and t :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> nat"   and s :: "'b" shows "On_Off.T step t s [] [] = 0"
quantale_hom_radj: fixes f :: "'b \<Rightarrow> 'c" assumes "f \<in> quantale_homset" shows "f \<stileturn> radj f"
lprefix_trans: fixes xs :: "'a llist"   and ys :: "'a llist"   and zs :: "'a llist" assumes "lprefix xs ys"   and "lprefix ys zs" shows "lprefix xs zs"
cases: fixes x :: "('v \<times> 'v \<Rightarrow> ennreal) \<times> 'v vertex \<times> 'v vertex" obtains f :: "'v \<times> 'v \<Rightarrow> ennreal" and xa :: "'v" and x' :: "'v" and y :: "'v" where "x = (fa__, Vertex xa__, Edge x' y__)"   | f :: "'v \<times> 'v \<Rightarrow> ennreal" and xa :: "'v" and y' :: "'v" and y :: "'v" where "x = (fb__, Edge xa y', Vertex y)"   | f :: "'v \<times> 'v \<Rightarrow> ennreal" and vb :: "'v" and vc :: "'v" and v :: "'v" and vd :: "'v" where "x = (fc__, Edge vb__ vc, Edge v vd)"   | f :: "'v \<times> 'v \<Rightarrow> ennreal" and va :: "'v" and vb :: "'v" where "x = (f, Vertex va, Vertex vb)"
last_snoc: fixes xs :: "'d list"   and x :: "'d" shows "last (xs @ [x]) = x"
simps_2: fixes cs :: "(Term_Class.term \<times> pterm) fset" shows "pre_strong_term_class.wellformed (Pabs cs) = (fBall cs (\<lambda>(pat, t). linear pat \<and> pre_strong_term_class.wellformed t) \<and> is_fmap cs \<and> pattern_compatibles cs \<and> cs \<noteq> {||})"
mult_left_less_imp_less: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_semiring plus minus less_eq less zero times"   and "less (times c a) (times c b)"   and "less_eq zero c" shows "less a b"
r_hd_def: shows "r_hd \<equiv> Cn 1 r_ifz [Cn 1 r_pdec1 [r_dec], Cn 1 r_pdec2 [r_dec], Cn 1 r_pdec12 [r_dec]]"
left_treeI: fixes u :: "nat"   and s :: "nat"   and t :: "nat" assumes "u \<in> V"   and "s \<in> set (u \<leadsto> t)" shows "u \<in> left_tree s t"
consistent_not_prv_not_prv_PPf: fixes \<phi> :: "'fmla"   and n :: "'trm" assumes "consistent"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "\<not> prv \<phi>"   and "n \<in> num" shows "\<not> bprv (PPf n \<langle>\<phi>\<rangle>)"
trans: fixes P :: "('a, 'b) Proc"   and Q :: "('a, 'b) Proc"   and Enc :: "'a \<Rightarrow> 'b"   and R :: "('a, 'b) Proc" assumes "(P, Q) \<in> encoding.indRelEQ Enc"   and "(Q, R) \<in> encoding.indRelEQ Enc" shows "(P, R) \<in> encoding.indRelEQ Enc"
hyps: fixes s :: "'a" assumes "\<Gamma>\<turnstile>sequence Seq (xs @ ys) \<down> Normal s" shows "\<Gamma>\<turnstile>sequence Seq xs \<down> Normal s \<and> (\<forall>s'. \<Gamma>\<turnstile> \<langle>sequence Seq xs,Normal s\<rangle> \<Rightarrow> s' \<longrightarrow> \<Gamma>\<turnstile>sequence Seq ys \<down> s')"
valid_propagate_2': fixes \<sigma> :: "bool"   and A :: "bool \<Rightarrow> ('b \<times> bool) option"   and M :: "'b \<Rightarrow> bool \<Rightarrow> (bool \<times> bool) option" assumes "\<sigma> \<Turnstile> bind\<^sub>S\<^sub>E A M" shows "\<exists>a. A \<sigma> = Some a \<and> snd a \<Turnstile> M (fst a)"
goal_noteval: shows "\<not> sinvar \<lparr>nodes = V, edges = E\<rparr> nP"
prime_elem_def: fixes p :: "'b" shows "prime_elem p = (p \<noteq> (0::'b) \<and> \<not> p dvd (1::'b) \<and> (\<forall>a b. p dvd a * b \<longrightarrow> p dvd a \<or> p dvd b))"
fractrel_def: shows "fractrel = (\<lambda>x y. snd x \<noteq> (0::'a) \<and> snd y \<noteq> (0::'a) \<and> fst x * snd y = fst y * snd x)"
euclidean_size_poly_def: fixes p :: "'b poly" shows "euclidean_size p = (if p = 0 then 0 else 2 ^ degree p)"
s_S_defs_2: shows "s' = \<lparr>inst = insts', funcs = fs', tab = clss', mem = bss', globs = gs'\<rparr>"
simps_2: fixes s :: "nat"   and ss :: "nat list"   and x :: "'c"   and y :: "'c"   and xs :: "'c list" shows "ALG_P (s # ss) x y xs = (if Suc s < length (swaps ss xs) then if swaps ss xs ! s = x \<and> swaps ss xs ! Suc s = y \<or> swaps ss xs ! s = y \<and> swaps ss xs ! Suc s = x then 1 else 0 else 0) + ALG_P ss x y xs"
mkNode_def: fixes c :: "'f"   and l :: "(('d \<times> 'e) \<times> 'f \<times> 'd \<times> 'e) tree"   and a :: "'d \<times> 'e"   and r :: "(('d \<times> 'e) \<times> 'f \<times> 'd \<times> 'e) tree" shows "mkNode c l a r \<equiv> \<langle>l, (a, c, min_kp a l r), r\<rangle>"
cond_def: fixes g :: "'g" shows "cond g \<equiv> \<lambda>(u, p). CFG_SSA_wf_base.redundant \<alpha>n inEdges' defs (uses(g := u)) (phis(g := p)) g"
not_reduced_word_for: fixes bs :: "'a list"   and A :: "'a set"   and a :: "'a"   and as :: "'a list" assumes "bs \<in> lists A"   and "sum_list bs = a"   and "order.greater (length as) (length bs)" shows "\<not> reduced_word_for A a as"
continuous_mult: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "continuous F f"   and "continuous F g" shows "continuous F (\<lambda>x. f x * g x)"
jointfun_def: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'c"   and m :: "nat"   and g :: "nat \<Rightarrow> 'c" shows "jointfun n f m g = (\<lambda>i. if i \<le> n then f i else g (sliden (Suc n) i))"
\<pi>_\<sigma>: fixes \<Sigma> :: "'b list"   and n :: "nat" shows "\<pi> ` (set \<circ> \<sigma> \<Sigma>) (n + 1) = (set \<circ> \<sigma> \<Sigma>) n"
setOfPairs: fixes pair :: "'c \<times> 'd set" shows "{(fst pair, {y}) |y. y \<in> snd pair} = {fst pair} \<times> {{y} |y. y \<in> snd pair}"
pos: fixes n :: "nat" shows "(0::'a) \<le> f n"
herbrand_evalt_1: fixes t :: "'a FOL_Fitting.term"   and e :: "nat \<Rightarrow> 'a hterm" assumes "closedt 0 t" shows "term_of_hterm (evalt e HApp t) = t"
abs_eq: fixes xb :: "nat"   and xa :: "int"   and x :: "int" shows "float_div_down xb xa x = float_of (real_div_down xb xa x)"
f14: shows "\<forall>x0 x1. (x1 \<le> x1 mod x0) = (x1 + - 1 * (x1 mod x0) \<le> 0)"
primitive_part_const_poly: fixes x :: "'a" shows "primitive_part [:x:] = [:unit_factor x:]"
assms: shows "<\<nu>x>a<y>.P \<longmapsto> \<alpha> \<prec> P'"
right_unique_iff: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "right_unique R = (\<forall>z. \<exists>\<^sub>\<le>\<^sub>1 x. R z x)"
vrange_atD: fixes b :: "V" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r" shows "\<exists>a\<in>elts (\<D>\<^sub>\<circ> r). r\<lparr>a\<rparr> = b"
induct: fixes P :: "(int \<times> int) \<times> int \<times> int \<Rightarrow> (int \<times> int) list \<Rightarrow> bool"   and a0 :: "(int \<times> int) \<times> int \<times> int"   and a1 :: "(int \<times> int) list" assumes "\<And>uu_. P uu_ []"   and "\<And>uv_ uw_. P uv_ [uw_]"   and "\<And>c\<^sub>0 c\<^sub>1 p\<^sub>0 v va. \<lbrakk>\<And>x. \<lbrakk>x = find_closest p\<^sub>0 (dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) (v # va); dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) \<le> dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case x of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))\<rbrakk> \<Longrightarrow> P (c\<^sub>0, c\<^sub>1) (v # va); \<And>x. \<lbrakk>x = find_closest p\<^sub>0 (dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) (v # va); \<not> dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) \<le> dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case x of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))\<rbrakk> \<Longrightarrow> P (p\<^sub>0, x) (v # va)\<rbrakk> \<Longrightarrow> P (c\<^sub>0, c\<^sub>1) (p\<^sub>0 # v # va)" shows "P a0 a1"
Ord_linear_lt: fixes k :: "V"   and l :: "V" assumes "Ord k"   and "Ord l" obtains "k < l"   | "k = l"   | "l < k"
expand_set_containment: fixes V :: "'b set"   and E :: "('b \<times> 'b) set"   and M :: "('b \<times> 'b) set"   and e :: "'b \<times> 'b" assumes "matching V E M"   and "e \<in> M" shows "e \<in> E"
distinct_140: fixes x13 :: "Syntax.term"   and x5 :: "Syntax.term" shows "Roll x13 \<noteq> Inj1 x5"
int_polyE: fixes p :: "'a poly" assumes "\<And>n. poly.coeff p n \<in> \<int>" obtains p' :: "int poly" where "p = of_int_poly p'"
two_composable_arrows_of_spans_axioms_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data"   and \<nu> :: "'b arrow_of_spans_data" shows "two_composable_arrows_of_spans_axioms C \<mu> \<nu> \<equiv> span_vertical_category.src C \<mu> = span_vertical_category.trg C \<nu>"
genby_lcoset_refl: fixes a :: "'a"   and S :: "'a set" shows "a \<in> a +o \<langle>S\<rangle>"
prFresh_def: fixes g :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> 'c"   and SEM :: "('d, 'e, 'f, 'sort, 'g, 'b, 'h) semDom_scheme" shows "prFresh g SEM \<equiv> \<forall>ys y s X val val'. wls s X \<and> fresh ys y X \<and> sWlsVal SEM val \<and> sWlsVal SEM val' \<and> eqBut val val' ys y \<longrightarrow> g X val = g X val'"
exp_sum: fixes I :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "finite I" shows "exp (sum f I) = (\<Prod>x\<in>I. exp (f x))"
C1_diff_imp_diff: fixes f :: "real \<Rightarrow> 'a"   and S :: "real set" assumes "f C1_differentiable_on S" shows "f differentiable_on S"
t: shows "\<forall>t'. t \<noteq> Abrupt t'"
inf_commute: fixes x :: "'a"   and y :: "'a" shows "inf_class.inf x y = inf_class.inf y x"