hcju/isabelleps huggingface dataset:
hyps_3: assumes "\<And>a. a \<in> F_ \<Longrightarrow> b dvd f a" shows "sum f F_ div b = (\<Sum>a\<in>F_. f a div b)"
R0_def: shows "R0 = {(V, V'). \<exists>x e e'. V = [x:=e] \<and> V' = [x:=e'] \<and> e \<equiv>\<^bsub>DA x\<^esub> e'}"
check_in_heap_is_OK: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap"   and f :: "unit \<Rightarrow> ((_) heap, exception, 'n) prog" assumes "ptr |\<in>| object_ptr_kinds h" shows "h \<turnstile> ok Heap_Error_Monad.bind (check_in_heap ptr) f = h \<turnstile> ok f ()"
simps_16: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'b"   and f3 :: "'c \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'b"   and f4 :: "('c, 'd) Combinators \<Rightarrow> ('c, 'd) Combinators \<Rightarrow> 'b" shows "(case DenyAll of DenyAll \<Rightarrow> f1 | DenyAllFromTo x xa \<Rightarrow> f2 x xa | AllowPortFromTo x xa xb \<Rightarrow> f3 x xa xb | x \<oplus> xa \<Rightarrow> f4 x xa) = f1"
that_5: fixes b\<^sub>1 :: "bool" assumes "x = B\<cdot>(Discr b\<^sub>1)"   and "y = \<bottom>" shows "thesis"
tauLaw4CongSimLeft: fixes Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<Psi> :: "'b"   and \<alpha> :: "'a prefix"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "\<And>\<Psi> P. (\<Psi>, P, P) \<in> Rel"   and "\<And>\<Psi> P Q. \<Psi> \<rhd> P \<sim> Q \<Longrightarrow> (\<Psi>, P, Q) \<in> Rel" shows "\<Psi> \<rhd> (\<alpha>\<cdot>P) \<oplus>\<^sub>\<top> \<alpha>\<cdot>(\<tau>.P) \<oplus>\<^sub>\<top> Q \<leadsto>\<guillemotleft>Rel\<guillemotright> \<alpha>\<cdot>(\<tau>.P) \<oplus>\<^sub>\<top> Q"
gcd_diff2: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and n :: "'b"   and m :: "'b" assumes "class.ring_gcd one zero times gcd lcm divide plus minus unit_factor normalize uminus" shows "gcd (minus n m) n = gcd m n"
inducts: fixes A :: "'a \<Rightarrow> bool"   and x :: "'a list"   and P :: "'a list \<Rightarrow> bool" assumes "listsp A x"   and "P []"   and "\<And>a l. \<lbrakk>A a; listsp A l; P l\<rbrakk> \<Longrightarrow> P (a # l)" shows "P x"
continuous_on_chart_inv: fixes s :: "'c set"   and c :: "('e, 'd) chart"   and f :: "'c \<Rightarrow> 'e" assumes "continuous_on s (apply_chart c \<circ> f)"   and "f ` s \<subseteq> domain c" shows "continuous_on s f"
P1: shows "C InAngle A B Q \<and> A B Q CongA D E F"
multiplicity_radical_prime: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b"   and x :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p"   and "x \<noteq> zero" shows "factorial_semiring.multiplicity times one p (factorial_semiring.radical zero times one normalize x) = (if dvd.dvd times p x then 1 else 0)"
ClassCast_def: shows "ClassCast \<equiv> ''ClassCast''"
sum_le_suminf: fixes f :: "nat \<Rightarrow> 'b"   and I :: "nat set" assumes "summable f"   and "finite I"   and "\<And>n. n \<in> - I \<Longrightarrow> (0::'b) \<le> f n" shows "sum f I \<le> suminf f"
zip_eq_Nil_iff: fixes xs :: "'b list"   and ys :: "'c list" shows "(xs || ys = []) = (xs = [] \<or> ys = [])"
expand_big_forest: fixes H :: "'a"   and d :: "'a" assumes "stone_kleene_relation_algebra.big_forest (\<lambda>x. top) top (\<sqinter>) (\<le>) (\<sqinter>) id uminus H d" shows "top \<sqinter> top = top \<squnion> top"
Red_I_of_subset_F: fixes N :: "'f set"   and N' :: "'f set" assumes "N \<subseteq> N'" shows "Red_I_\<G> N \<subseteq> Red_I_\<G> N'"
relabel_measure: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and u :: "nat" assumes "Height_Bounded_Labeling c s t f l"   and "Network.relabel_precond c t f l u" shows "Network.sum_heights_measure c (Network.relabel_effect c f l u) < Network.sum_heights_measure c l"
rec: shows "\<forall>n>n\<^sub>0. f n = f (nat \<lceil>a * real n + b\<rceil>) + f (nat \<lceil>c * real n + d\<rceil>) + C\<^sub>1 * real n + C\<^sub>2"
rpr_less_imp_le: fixes x :: "'a"   and r :: "('a \<times> 'a) set"   and y :: "'a" assumes "x \<^bsub>r\<^esub>\<prec> y" shows "x \<^bsub>r\<^esub>\<preceq> y"
distinct_7: fixes x1 :: "name"   and x51 :: "term"   and x52 :: "term" shows "Const x1 \<noteq> x51 $ x52"
SD_implies_SR: assumes "SD \<V> Tr\<^bsub>(induceES SES)\<^esub>" shows "SR \<V> Tr\<^bsub>(induceES SES)\<^esub>"
full_gb_isGB: fixes bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" shows "is_Groebner_basis (fst ` set (full_gb bs))"
measurable_gaussian: shows "gaussian \<in> stock_measure (PRODUCT REAL REAL) \<rightarrow>\<^sub>M subprob_algebra (stock_measure REAL)"
NatTransFtor2: fixes \<eta> :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" assumes "NatTransP \<eta>" shows "Functor (NTCod \<eta>)"
b2: fixes s1 :: "lam"   and s2 :: "lam"   and t :: "lam" assumes " s1 \<longrightarrow>\<^sub>\<beta> s2" shows " App t s1 \<longrightarrow>\<^sub>\<beta> App t s2"
distinct': fixes a :: "'a"   and b1 :: "nat"   and b2 :: "nat" assumes "(a, b1) \<in> set cs"   and "(a, b2) \<in> set cs" shows "b1 = b2"
1: shows "B1 \<noteq> 0 \<and> 0 < arg B1 \<longrightarrow> \<not> 0 < arg (- B1)"
C_eq_rd: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "p \<noteq> []" shows "Cp (list2FWpolicy (remdups p)) = Cp (list2FWpolicy p)"
is_iso_dghmI: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "v11 (\<FF>\<lparr>ObjMap\<rparr>)"   and "v11 (\<FF>\<lparr>ArrMap\<rparr>)"   and "\<R>\<^sub>\<circ> (\<FF>\<lparr>ObjMap\<rparr>) = \<BB>\<lparr>Obj\<rparr>"   and "\<R>\<^sub>\<circ> (\<FF>\<lparr>ArrMap\<rparr>) = \<BB>\<lparr>Arr\<rparr>" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>i\<^sub>s\<^sub>o\<^bsub>\<alpha>\<^esub> \<BB>"
under_Under_singl: fixes r :: "('a \<times> 'a) set"   and a :: "'a" shows "under r a = Under r {a}"
intersecttwo: fixes X :: "'b set set"   and \<A> :: "'b set set set"   and A :: "'b set set"   and A' :: "'b set set"   and x :: "'b set"   and C :: "'b set" assumes "ChamberComplexWithApartmentSystem X \<A>"   and "A \<in> \<A>"   and "A' \<in> \<A>"   and "x \<in> A \<inter> A'"   and "C \<in> A \<inter> A'"   and "SimplicialComplex.maxsimp X C" shows "\<exists>f. ChamberComplexIsomorphism A A' f \<and> fixespointwise f x \<and> fixespointwise f C"
eq0: shows "z ^ 3 + a * z\<^sup>2 + b * z + c = 0"
neg_less_pos: fixes 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.linordered_ab_group_add plus zero minus uminus less_eq less" shows "less (uminus a) a = less zero a"
norm_vec_def: fixes x :: "('b, 'c) vec" shows "norm x = L2_set (\<lambda>i. norm (x $ i)) UNIV"
call_instr_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = snd (fst (call_instr instr s1)) \<and> t2 = snd (fst (call_instr instr s2))" shows "low_equal t1 t2"
free_unfree: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and u :: "'c"   and name :: "name" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree"   and "unfree u = Some name" shows "u = free name"
thread_oks_new_thread: fixes ts :: "'a \<Rightarrow> ('b \<times> 'c \<Rightarrow>f nat) option"   and tas :: "('a, 'd, 'e) new_thread_action list"   and t :: "'a"   and x :: "'d"   and m :: "'e" assumes "thread_oks ts tas"   and "NewThread t x m \<in> set tas" shows "ts t = None"
rk_couple_to_singleton: fixes A :: "Points"   and B :: "Points" assumes "rk {A, B} = 1" shows "A = B"
incseqD: fixes f :: "nat \<Rightarrow> 'a"   and i :: "nat"   and j :: "nat" assumes "incseq f"   and "i \<le> j" shows "f i \<le> f j"
DFA_def: shows "DFA = \<lparr>states = h ` UNIV // R, init = h (R `` {[]}), final = {h (R `` {u}) |u. u \<in> A}, nxt = \<lambda>q x. h (\<Union>u\<in>h\<^sup>-\<^sup>1 q. R `` {u @ [x]})\<rparr>"
rep_eq: fixes x :: "nat"   and xa :: "nat_int" shows "(x \<^bold>\<notin> xa) = (x \<notin> Rep_nat_int xa)"
1: shows "simple_match_to_of_match_single r a b c d = simple_match_to_of_match_single r f g h i"
monic_prod_mset: fixes as :: "'a poly multiset" assumes "\<And>a. a \<in># as \<Longrightarrow> monic a" shows "monic (\<Prod>\<^sub># as)"
gbinomial_int_Suc_Suc: fixes x :: "int"   and k :: "nat" shows "x + 1 gchoose Suc k = x gchoose k + (x gchoose Suc k)"
d_star_def: shows "d\<^sup>* \<equiv> Max (card ` S ` {1..m})"
iless_eSuc0: fixes n :: "enat" shows "(n < eSuc 0) = (n = 0)"
correct_jvm_state_def: fixes \<Phi> :: "String.literal \<Rightarrow> String.literal \<Rightarrow> (ty list \<times> ty err list) option list" shows "correct_jvm_state \<Phi> = {s. correct_state_ts \<Phi> (thr s) (shr s) \<and> lock_thread_ok (locks s) (thr s)}"
fresh_fact: fixes z :: "name"   and s :: "lam"   and y :: "name"   and t :: "lam" assumes "z \<sharp> s"   and "z = y \<or> z \<sharp> t" shows "z \<sharp> t[y::=s]"
hyps_1: fixes x :: "('b option, 'a, 'c option) gpv" shows "interaction_bound' x \<le> interaction_bound consider x"
weight_le: shows "weight_spmf p \<le> weight_spmf q"
algebraicE': fixes x :: "'a" assumes "algebraic x" obtains p :: "int poly" where "p \<noteq> 0" and "poly (of_int_poly p) x = (0::'a)"
RModule_def: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set" shows "RModule R smult M \<equiv> R_scalar_mult R \<and> Group M \<and> RModule_axioms R smult M"
alg_closure_def: shows "alg_closure = (SOME L. algebraic_closure L (indexed_const ` carrier R) \<and> indexed_const \<in> ring_hom R L)"
simps: fixes d :: "nat"   and qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and B :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) set" shows "shift1_inv d (qs, B) = (B = {q \<in> set qs. poly_deg (fst q) = d \<and> m < card (snd q)} \<and> shift2_inv d qs)"
create_element_is_weakly_dom_component_safe: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and tag :: "char list"   and result :: "(_) element_ptr"   and h' :: "(_) heap" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> create_element document_ptr tag \<rightarrow>\<^sub>r result"   and "h \<turnstile> create_element document_ptr tag \<rightarrow>\<^sub>h h'" shows "is_weakly_dom_component_safe {cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^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 document_ptr} {cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^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 result} h h'"
struct_maps_def: fixes T :: "('e, 'g) partial_object_scheme"   and S :: "('f, 'h) partial_object_scheme" shows "struct_maps T S = {f \<in> carrier T \<rightarrow> carrier S. f = restrict f (carrier T)}"
set_card: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a" shows "|local.set a| <o natLeq"
WTrtNil: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option" shows "P,E,h,sh \<turnstile> [] [:] []"
infsetsum_cong: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and B :: "'a set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x"   and "A = B" shows "infsetsum f A = infsetsum g B"
INF_lower: fixes Inf :: "'a::type set \<Rightarrow> 'a::type"   and Sup :: "'a::type set \<Rightarrow> 'a::type"   and inf :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type"   and less_eq :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and less :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and sup :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type"   and bot :: "'a::type"   and top :: "'a::type"   and i :: "'b::type"   and A :: "'b::type set"   and f :: "'b::type \<Rightarrow> 'a::type" assumes "class.complete_lattice (Inf::'a::type set \<Rightarrow> 'a::type) (Sup::'a::type set \<Rightarrow> 'a::type) (inf::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (less::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (sup::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (bot::'a::type) (top::'a::type)"   and "(i::'b::type) \<in> (A::'b::type set)" shows "(less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) ((Inf::'a::type set \<Rightarrow> 'a::type) ((f::'b::type \<Rightarrow> 'a::type) ` (A::'b::type set))) (f (i::'b::type))"
map_try_spmf: fixes f :: "'i \<Rightarrow> 'h"   and p :: "'i spmf"   and q :: "'i spmf" shows "map_spmf f (TRY p ELSE q) = TRY map_spmf f p ELSE map_spmf f q"
tauLaw2StatImpLeft: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<And>\<Psi>''. (\<Psi> \<otimes> \<Psi>'', P \<oplus>\<^sub>\<top> \<tau>.P, \<tau>.P) \<in> Rel" shows "\<Psi> \<rhd> P \<oplus>\<^sub>\<top> \<tau>.P \<lessapprox><Rel> \<tau>.P"
null_def: fixes xs :: "'c list" shows "List.null xs = (xs = [])"
ccSUP_least: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'a"   and u :: "'a" assumes "countable A"   and "\<And>i. i \<in> A \<Longrightarrow> f i \<le> u" shows "Sup (f ` A) \<le> u"
reduction_inv_3: fixes u :: "db"   and t' :: "db"   and P :: "bool" assumes "DBAbsN u \<rightarrow>\<^sub>v t'" shows "P"
finite_SMP_representationD_1: fixes M :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list" assumes "finite_SMP_representation arity Ana \<Gamma> M" shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (set M)"
scopeExtSum: fixes x :: "name"   and P :: "pi"   and Q :: "pi" assumes "x \<sharp> P" shows "<\<nu>x>(P \<oplus> Q) \<simeq> P \<oplus> <\<nu>x>Q"
inject: fixes x1 :: "'a"   and y1 :: "'a" shows "(ln x1 = ln y1) = (x1 = y1)"
fmdom'_alt_def: fixes m :: "('a, 'b) fmap" shows "fmdom' m = fset (fmdom m)"
is_unit_left_imp_coprime: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a one" shows "algebraic_semidom.coprime times one a b"
Fun_mkArr: fixes A :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr set"   and B :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr set"   and F :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr \<Rightarrow> (('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr" assumes "S.arr (S.mkArr A B F)" shows "S.Fun (S.mkArr A B F) = restrict F A"
one_plus_numeral: fixes n :: "num" shows "(1::'a) + numeral n = numeral (num.One + n)"
list_induct3_snoc_Conssnoc_Cons_pairwise: fixes P :: "'c list \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'a list"   and zs :: "'b list" assumes "\<And>ys zs. P [] ys zs"   and "\<And>xs zs. P xs [] zs"   and "\<And>xs ys. P xs ys []"   and "\<And>xs x y z zs. P (xs @ [x]) [y] (z # zs)"   and "\<And>xs x y ys w z zs. \<lbrakk>P xs ys zs; P xs (ys @ [w]) (z # zs); P (xs @ [x]) (y # ys) zs\<rbrakk> \<Longrightarrow> P (xs @ [x]) (y # ys @ [w]) (z # zs)" shows "P xs ys zs"
PLam_def: fixes I :: "nat set"   and F :: "nat \<Rightarrow> ('c \<times> (nat \<Rightarrow> 'c) \<times> 'd) program" shows "PLam I F \<equiv> \<Squnion>i\<in>I. lift i (F i)"
is_median_def: fixes x :: "'c"   and xs :: "'c list" shows "is_median x xs = (length (filter (\<lambda>y. y < x) xs) \<le> length xs div 2 \<and> length (filter ((<) x) xs) \<le> length xs div 2)"
subdeg_sum'_1: shows "fls_subdegree f = - fls_subdegree g"
add_mset_remove_trivial_eq: fixes N :: "'a multiset"   and a :: "'a" shows "(N = add_mset a (N \<ominus> a)) = (a \<in># N)"
simps_3: fixes a :: "('b, 'c) Combinators"   and v :: "'b"   and va :: "'b" shows "member a (DenyAllFromTo v va) = (a = DenyAllFromTo v va)"
step_deterministic: fixes s1 :: "('fenv_std, 'henv, ('fun, 'dyn) frame) state"   and s2 :: "('fenv_std, 'henv, ('fun, 'dyn) frame) state"   and s3 :: "('fenv_std, 'henv, ('fun, 'dyn) frame) state" assumes "Sstd.step s1 s2"   and "Sstd.step s1 s3" shows "s2 = s3"
set_intros_3: fixes x3 :: "'a" shows "x3 \<in> set3_sum3 (In3 x3)"
hyps: fixes y :: "nat" assumes "y < s"   and "y \<in> S" shows "\<exists>n. wellorder_class.enumerate S n = y"
intros: fixes f :: "nat \<Rightarrow> 'd"   and g :: "nat \<Rightarrow> 'd" assumes "finite {n. f n \<noteq> g n}" shows "aeseq_eq f g"
lm152: fixes R :: "('c \<times> 'd) set"   and f :: "('c \<times> 'd) set" assumes "R \<subseteq> f"   and "runiq f"   and "Domain f \<subseteq> Domain R" shows "f = R"
disjoint_eq_subset_Compl: fixes A :: "'a set"   and B :: "'a set" shows "(A \<inter> B = {}) = (A \<subseteq> - B)"
lang_def: fixes G :: "('a, 'b, 'c) igba_rec_scheme" assumes "igba G" shows "Automata.igba.lang G \<equiv> Collect (Automata.igba.accept G)"
leastI2_order: fixes A :: "'d set"   and P :: "'d \<Rightarrow> bool" assumes "A \<noteq> {}"   and "\<And>k. \<lbrakk>k \<in> A; \<And>l. l \<in> A \<Longrightarrow> k \<le> l\<rbrakk> \<Longrightarrow> P k" shows "P (least A)"
comp_cod_arr: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'c"   and b :: "'c" assumes "category C"   and "partial_magma.arr C f"   and "partial_magma.cod C f = b" shows "C b f = f"
arctan_approx: fixes x :: "real"   and n :: "nat" assumes "0 \<le> x"   and "x < 1"   and "even n" shows "arctan x - arctan_approx n x \<in> {0..x ^ (2 * n + 1) / (1 - x ^ 4)}"
expand: fixes option :: "'c option"   and option' :: "'c option" assumes "(option = None) = (option' = None)"   and "\<lbrakk>option \<noteq> None; option' \<noteq> None\<rbrakk> \<Longrightarrow> the option = the option'" shows "option = option'"
lookup_update: fixes ks :: "'c list"   and v :: "'b"   and t :: "('c, 'b) trie"   and ks' :: "'c list" shows "lookup_trie (update_trie ks v t) ks' = (if ks = ks' then Some v else lookup_trie t ks')"
elem_index_def: assumes "class.countable TYPE('a)" shows "countable.elem_index = (SOME f. inj f)"
deriv_c1: shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P\<^sub>1 c\<^sub>1 Q,A"
terminates_in_def: fixes e :: "('r, 'l, 'v) expr"   and s' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" shows "e \<down> s' \<equiv> \<exists>s. maximal_execution e s s'"
homotopic_paths_assoc: fixes p :: "real \<Rightarrow> 'b"   and s :: "'b set"   and q :: "real \<Rightarrow> 'b"   and r :: "real \<Rightarrow> 'b" assumes "path p"   and "path_image p \<subseteq> s"   and "path q"   and "path_image q \<subseteq> s"   and "path r"   and "path_image r \<subseteq> s"   and "pathfinish p = pathstart q"   and "pathfinish q = pathstart r" shows "homotopic_paths s (p +++ q +++ r) ((p +++ q) +++ r)"
aff_dim_inner_basis_exists: fixes V :: "'b set" shows "\<exists>B\<subseteq>V. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> int (card B) = aff_dim V + 1"
hull_subset: fixes s :: "'a set"   and S :: "'a set \<Rightarrow> bool" shows "s \<subseteq> S hull s"
forks_transitions_def: fixes n :: "nat"   and fs :: "nat list" shows "Tr\<^sub>F n fs = (\<Union>i<n. Tr\<^sub>f i (fs ! i))"
1: assumes "\<A> \<Turnstile> tseytin_toatom F F"   and "\<A> \<Turnstile> F" shows "tseytin_update \<A> F \<Turnstile> tseytin_toatom F F"
comp_null_1: fixes f :: "'a" shows "C.null \<cdot>\<^sub>C f = C.null"
HFail_valueChosen2: fixes s :: "state"   and s' :: "state"   and q :: "Proc"   and D :: "Disk set"   and b :: "nat"   and p :: "Proc" assumes "HFail s s' q"   and "\<forall>d\<in>D. b \<le> bal (disk s d p) \<and> (\<forall>q. phase s q = 1 \<and> b \<le> mbal (dblock s q) \<and> hasRead s q d p \<longrightarrow> (\<exists>br\<in>blocksRead s q d. b \<le> bal (block br)))" shows "\<forall>d\<in>D. b \<le> bal (disk s' d p) \<and> (\<forall>q. phase s' q = 1 \<and> b \<le> mbal (dblock s' q) \<and> hasRead s' q d p \<longrightarrow> (\<exists>br\<in>blocksRead s' q d. b \<le> bal (block br)))"
prefix_not_empty_length_gt_zero: fixes xs :: "'b list"   and ys :: "'b list" assumes "strict_prefix xs ys"   and "xs \<noteq> []" shows "0 < length ys"
maxchain_def: fixes A :: "'c set set"   and C :: "'c set set" shows "subset.maxchain A C = (subset.chain A C \<and> (\<nexists>S. subset.chain A S \<and> C \<subset> S))"
double_of_int_def: fixes x :: "int" shows "double_of_int x = double_of_integer (integer_of_int x)"
n_2: shows "ldrop (enat x) (lfilter P lst) = lfilter P (ldrop (enat n) lst)"
simps_3: fixes \<sigma> :: "'tp" shows "TE_resOf (Wit \<sigma>) = \<sigma>"
comp_Suc: fixes n :: "nat"   and p :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "swapidseq n p"   and "a \<noteq> b" shows "swapidseq (Suc n) (Fun.swap a b id \<circ> p)"
And_assign: fixes \<phi> :: "formula"   and \<psi> :: "formula" assumes "safe_formula \<phi>"   and "safe_assignment (fv \<phi>) \<psi>"   and "P \<phi>" shows "P (And \<phi> \<psi>)"
SKIP_SEQ: fixes P :: "'a process" shows "(SKIP `;` P) = P"
assms: shows "\<forall>x\<in>set xs. bound0 (f x)"
NEQ_quot_tm: fixes i :: "name"   and j :: "name" assumes "i \<noteq> j" shows "{} \<turnstile> \<guillemotleft>Var i\<guillemotright> NEQ \<guillemotleft>Var j\<guillemotright>"
hyps: shows "p \<noteq> [] \<longrightarrow> C (list2FWpolicy p) x = C (list2FWpolicy (removeAllDuplicates p)) x"
thetaWhileZOT_sym: shows "sym thetaWhileZOT"
in_idem: fixes A :: "'c DBMEntry set"   and x :: "'c DBMEntry" assumes "finite A"   and "x \<in> A" shows "linordered_monoid.min x (linordered_monoid.Min A) = linordered_monoid.Min A"
assms_3: shows "A \<noteq> []\<^sub>\<circ>"
assms_1: shows "a \<in>\<^sub>\<circ> r -`\<^sub>\<bullet> A"
node_deliverTE'': fixes ii :: "nat"   and s :: "'c"   and R :: "nat set"   and i :: "nat"   and d :: "nat"   and s' :: "'c net_state"   and S :: "('c \<times> 'd seq_action \<times> 'c) set" assumes "(NodeS ii s R, i:deliver(d), s') \<in> node_sos S" obtains s' :: "'c" where "i = ii" and "s'b__ = NodeS ii s' R" and "(s, deliver d, s') \<in> S"