hcju/isabelleps huggingface dataset:
sibling_swapSyms_other_sibling: fixes t :: "'a tree"   and b :: "'a"   and a :: "'a" assumes "consistent t"   and "sibling t b \<noteq> a"   and "sibling t b \<noteq> b"   and "a \<noteq> b" shows "sibling (swapSyms t a b) (sibling t b) = a"
success_run: fixes f :: "'b Heap"   and h :: "heap" assumes "success f h" obtains h' :: "heap" and r :: "'b" where "run f (Some h) (Some h') r"
map_id: fixes t :: "('d, 'e, 'f) F2" shows "F2map id id id t = t"
nth_rows: fixes i :: "nat"   and A :: "'b mat" assumes "i < dim_row A" shows "rows A ! i = row A i"
Inter_lower: fixes B :: "'c set"   and A :: "'c set set" assumes "s2p A B" shows "\<Inter> A \<subseteq> B"
fun_eq_on_def: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and A :: "'c set" shows "fun_eq_on f g A \<equiv> \<forall>a\<in>A. f a = g a"
backward_lockstep_simulation: fixes s2 :: "('fenv_inca, 'henv, ('fun, 'dyn) frame) state"   and s2' :: "('fenv_inca, 'henv, ('fun, 'dyn) frame) state"   and s1 :: "('fenv_std, 'henv, ('fun, 'dyn) frame) state" assumes "Sinca.step s2 s2'"   and "s1 \<sim> s2" shows "\<exists>s1'. Sstd.step s1 s1' \<and> s1' \<sim> s2'"
composite_functor_def: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'a \<Rightarrow> 'c"   and G :: "'c \<Rightarrow> 'd" shows "composite_functor A B C F G \<equiv> functor A B F \<and> functor B C G"
inf_sup_ord_1: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.lattice inf less_eq less sup" shows "less_eq (inf x y) x"
Abs_finfun_inject: fixes x :: "'d \<Rightarrow> 'e"   and y :: "'d \<Rightarrow> 'e" assumes "x \<in> finfun"   and "y \<in> finfun" shows "(Abs_finfun x = Abs_finfun y) = (x = y)"
simps_109: fixes x91 :: "'f"   and x92 :: "'d set"   and x93 :: "('d, 'e, 'f) com"   and x8 :: "'d \<Rightarrow> ('d, 'e, 'f) com" shows "Guard x91 x92 x93 \<noteq> DynCom x8"
exp_Euler: fixes z :: "complex" shows "exp (\<i> * z) = cos z + \<i> * sin z"
memberid_mdecl_simp: fixes sig :: "sig"   and m :: "methd" shows "memberid (mdecl (sig, m)) = mid sig"
map_of_is_SomeI: fixes xys :: "('b \<times> 'c) list"   and x :: "'b"   and y :: "'c" assumes "distinct (map fst xys)"   and "(x, y) \<in> set xys" shows "map_of xys x = Some y"
NRO_sender: fixes A' :: "agent"   and B :: "agent"   and n :: "msg"   and b :: "msg"   and l :: "msg"   and C :: "msg"   and A :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Says A' B \<lbrace>n, b, l, C, Crypt (priEK A) X\<rbrace> \<in> set evs"   and "evs \<in> zg" shows "A' \<in> {A, Spy}"
space_point_measure: fixes A :: "'a set"   and f :: "'a \<Rightarrow> ennreal" shows "space (point_measure A f) = A"
is_up_def: shows "is_up \<equiv> map_fun (map_fun Word.Word Word.rep) id (\<lambda>_. LENGTH('b) \<le> LENGTH('c))"
lg_inv: fixes x :: "nat" assumes "0 < x" shows "lg (2 ^ x) = x"
p_def: shows "path AB p q1 \<and> target p q1 = q2 \<and> distinct (q1 # states p q1)"
isSPath_pathLE: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and p :: "(nat \<times> nat) list"   and t :: "nat" assumes "Graph.isPath c s p t" shows "\<exists>p'. Graph.isSimplePath c s p' t"
nt_abbrevE: fixes f :: "('d, 'e, 'f, 'g, 'h, 'i) NatTrans"   and F :: "('d, 'e, 'f, 'g, 'h, 'i) Functor"   and G :: "('d, 'e, 'f, 'g, 'h, 'i) Functor" assumes "NT f : F \<Longrightarrow> G" obtains "NatTrans f" and "NTDom f = F" and "NTCod f = G"
cod_pr0: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "cod \<pp>\<^sub>0[a, b] = b"
dom_trms_def: fixes C :: "'a literal set"   and E :: "'a trm set" shows "dom_trms C E = {x \<in> E. dom_trm x C}"
except_except: fixes p :: "'b \<Rightarrow>\<^sub>0 'c"   and S :: "'b set"   and T :: "'b set" shows "except (except p S) T = except p (S \<union> T)"
onl_def: fixes \<Gamma> :: "'d \<Rightarrow> ('b, 'c, 'd, 'e) seqp"   and P :: "'a \<times> 'e \<Rightarrow> bool" shows "onl \<Gamma> P \<equiv> \<lambda>(\<xi>, p). \<forall>l\<in>labels \<Gamma> p. P (\<xi>, l)"
single_Diff_lessThan: fixes k :: "'b" shows "\<eta> k - {..<k} = \<eta> k"
quot_mem: fixes X :: "'a set" assumes "X \<in> carrier (R Quot a_kernel R S h)" shows "\<exists>x\<in>carrier R. X = a_kernel R S h +> x"
coreflexive_pseudo_complement: fixes uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and conv :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.pd_allegory uminus inf less_eq less sup bot top one times conv"   and "less_eq x one" shows "(inf x y = bot) = less_eq x (inf (uminus y) one)"
Derives1_sentence2: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and a :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and b :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "CFG.Derives1 \<NN> \<TT> \<RR> a i r b" shows "CFG.is_sentence \<NN> \<TT> b"
assms_1: shows "pseudominimal_in (P .\<^bold>\<le> x) w"
heap_wellformed_induct_rev_si: fixes h :: "(_) heap"   and P :: "(_) object_ptr \<Rightarrow> bool"   and ptr :: "(_) object_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "\<And>child. \<lbrakk>\<And>parent child_node. \<lbrakk>child = cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^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 child_node; h \<turnstile> Shadow_DOM.get_parent child_node \<rightarrow>\<^sub>r Some parent\<rbrakk> \<Longrightarrow> P parent; \<And>host shadow_root. \<lbrakk>child = 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; h \<turnstile> get_host shadow_root \<rightarrow>\<^sub>r host\<rbrakk> \<Longrightarrow> P (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 host); \<And>disc_doc disc_node. \<lbrakk>child = cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^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 disc_node; h \<turnstile> get_disconnected_document disc_node \<rightarrow>\<^sub>r disc_doc\<rbrakk> \<Longrightarrow> P (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 disc_doc)\<rbrakk> \<Longrightarrow> P child" shows "P ptr"
Comp_Id_Hom: fixes B :: "'o"   and f :: "'a"   and A :: "'o" assumes "B \<in> Obj"   and "f \<in> Hom A B" shows "Comp B B A (Id B) f = f"
rel_mset_split: fixes rel :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and X1 :: "'a multiset"   and X2 :: "'a multiset"   and Y :: "'b multiset" assumes "rel_mset rel (X1 + X2) Y" shows "\<exists>Y1 Y2. Y = Y1 + Y2 \<and> rel_mset rel X1 Y1 \<and> rel_mset rel X2 Y2"
M3: shows "Min {j. j < length a \<and> is_nonterminal (a ! j)} \<le> i"
small_big_mult: fixes f1 :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g1 :: "'a \<Rightarrow> 'b"   and f2 :: "'a \<Rightarrow> 'b"   and g2 :: "'a \<Rightarrow> 'b" assumes "f1 \<in> l F g1"   and "f2 \<in> L F g2" shows "(\<lambda>x. f1 x * f2 x) \<in> l F (\<lambda>x. g1 x * g2 x)"
sds_lowering_sd_efficient_def: fixes agents :: "'a set"   and alts :: "'b set"   and sds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b pmf"   and agents' :: "'a set"   and alts' :: "'b set" shows "sds_lowering_sd_efficient agents alts sds agents' alts' \<equiv> sd_efficient_sds agents alts sds \<and> sds_lowering agents alts sds agents' alts'"
fminusD1: fixes c :: "'c"   and A :: "'c fset"   and B :: "'c fset" assumes "c |\<in>| A |-| B" shows "c |\<in>| A"
accessible_def: shows "MN.accessible \<equiv> {q. MN.left_lang q \<noteq> {}}"
eval_def: fixes rcs :: "real \<times> real list"   and xs :: "real list" shows "eval rcs xs = fst rcs + \<langle>snd rcs,xs\<rangle>"
verit_sum_simplify: fixes a :: "'f" shows "a + (0::'f) = a"
and_list_semantic: fixes w :: "nat \<Rightarrow> 'a set"   and xs :: "'a ltln list" shows "w \<Turnstile>\<^sub>n and_list xs = (\<forall>x\<in>set xs. w \<Turnstile>\<^sub>n x)"
replicate_eq_appendE: fixes xs :: "'a list"   and ys :: "'a list"   and n :: "nat"   and x :: "'a" assumes "xs @ ys = replicate n x" obtains i :: "nat" and j :: "nat" where "n = i + j" and "xs = replicate i x" and "ys = replicate j x"
map_le_def: fixes m\<^sub>1 :: "'c \<Rightarrow> 'd option"   and m\<^sub>2 :: "'c \<Rightarrow> 'd option" shows "(m\<^sub>1 \<subseteq>\<^sub>m m\<^sub>2) = (\<forall>a\<in>dom m\<^sub>1. m\<^sub>1 a = m\<^sub>2 a)"
perm_fresh_fresh: fixes a :: "name"   and x :: "'d"   and b :: "name" assumes "a \<sharp> x"   and "b \<sharp> x" shows "[(a, b)] \<bullet> x = x"
subst_closed_except_id: fixes size :: "'a \<Rightarrow> nat"   and frees :: "'a \<Rightarrow> name fset"   and subst :: "'a \<Rightarrow> (name, 'a) fmap \<Rightarrow> 'a"   and "consts" :: "'a \<Rightarrow> name fset"   and app :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unapp :: "'a \<Rightarrow> ('a \<times> 'a) option"   and const :: "name \<Rightarrow> 'a"   and unconst :: "'a \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'a"   and unfree :: "'a \<Rightarrow> name option"   and abs_pred :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"   and t :: "'a"   and S :: "name fset"   and env :: "(name, 'a) fmap" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred"   and "pre_term.closed_except frees t S"   and "fdisjnt (fmdom env) S" shows "subst t env = t"
max_ideal_axioms: shows "max_ideal R I (+) (\<cdot>) \<zero> \<one>"
not_coprime_imp_witness: assumes "\<not> coprime a n" shows "local.fermat_witness"
subst_cls_lists_single: fixes C :: "'a literal multiset"   and \<sigma> :: "'s" shows "[C] \<cdot>\<cdot>cl [\<sigma>] = [C \<cdot> \<sigma>]"
interval_lebesgue_integral_def: fixes M :: "real measure"   and a :: "ereal"   and b :: "ereal"   and f :: "real \<Rightarrow> 'b" shows "interval_lebesgue_integral M a b f = (if a \<le> b then set_lebesgue_integral M (einterval a b) f else - set_lebesgue_integral M (einterval b a) f)"
simps_3: fixes x41 :: "bexp"   and x42 :: "com"   and x43 :: "com"   and y41 :: "bexp"   and y42 :: "com"   and y43 :: "com" shows "(IF x41 THEN x42 ELSE x43 = IF y41 THEN y42 ELSE y43) = (x41 = y41 \<and> x42 = y42 \<and> x43 = y43)"
range_mapping_Max_mapping: fixes X :: "'a set"   and i :: "nat"   and j :: "'a \<Rightarrow> nat"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> nat option" assumes "finite X"   and "X \<noteq> {}"   and "\<forall>x\<in>X. range_mapping i (j x) (P x)" shows "range_mapping i (Max (j ` X)) (Max_mapping (P ` X))"
cases: fixes x :: "'a ns_constraint" obtains p :: "linear_poly" and c :: "'a" where "x = LEQ_ns p c"   | v :: "linear_poly" and va :: "'a" where "x = GEQ_ns v va"
ok_sym: fixes F :: "'b program"   and G :: "'b program" assumes "F ok G" shows "G ok F"
closed_vimage: fixes s :: "'a set"   and f :: "'b \<Rightarrow> 'a" assumes "closed s"   and "continuous_on UNIV f" shows "closed (f -` s)"
gh: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and h :: "'b" assumes "three_composable_identities_in_bicategory_of_spans V H \<a> \<i> src trg f g h" shows "src g = trg h"
root_shorter: fixes u :: "'b list"   and r :: "'b list" assumes "u \<noteq> \<epsilon>"   and "u \<in> r*"   and "u \<noteq> r" shows "\<^bold>|r\<^bold>| < \<^bold>|u\<^bold>|"
intros: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Bs :: "base list"   and rest :: "(char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list"   and D :: "char list" assumes "class P C = \<lfloor>(Bs, rest)\<rfloor>"   and "Shares D \<in> set Bs" shows "subclsSp P C D"
interchange_rows_i: fixes A :: "(('a, 'b) vec, 'c) vec"   and i :: "'c"   and j :: "'c" shows "interchange_rows A i j $ i = A $ j"
paths_tri: fixes ab :: "'a set"   and a :: "'a"   and b :: "'a"   and bc :: "'a set"   and c :: "'a"   and ca :: "'a set" assumes "path ab a b"   and "path bc b c"   and "path ca c a"   and "a \<notin> bc" shows "\<triangle> a b c"
pmf_positive_iff: fixes p :: "'c pmf"   and x :: "'c" shows "(0 < pmf p x) = (x \<in> set_pmf p)"
2: shows "min_height \<langle>l, x, r\<rangle> = nat \<lfloor>log 2 (real (size1 r))\<rfloor> + 1"
split_comp_eq_pair: fixes a :: "'c" shows "(\<lambda>x. fst x = a) = (\<lambda>(x, y). x = a)"
map_simp_ide: fixes a :: "'a" assumes "D.J.ide a" shows "\<tau>o\<chi>.map a = \<tau> a \<cdot> \<chi> a"
unity_root_0_n: fixes n :: "int" shows "unity_root 0 n = 1"
max_intervalI: fixes a :: "'a"   and A :: "'a interval"   and b :: "'a"   and B :: "'a interval" assumes "a \<in>\<^sub>i A"   and "b \<in>\<^sub>i B" shows "max a b \<in>\<^sub>i max_interval A B"
lholed_same_type: fixes k :: "nat"   and lholed :: "Lholed"   and es :: "e list"   and les :: "e list"   and es' :: "e list"   and les' :: "e list"   and \<S> :: "s_context"   and \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list" assumes "Lfilled k lholed es les"   and "Lfilled k lholed es' les'"   and "\<S>\<bullet>\<C> \<turnstile> les : ts _> ts'"   and "\<And>arb_labs ts ts'. \<S>\<bullet>\<C>\<lparr>label := arb_labs @ label \<C>\<rparr> \<turnstile> es : ts _> ts' \<Longrightarrow> \<S>\<bullet>\<C>\<lparr>label := arb_labs @ label \<C>\<rparr> \<turnstile> es' : ts _> ts'" shows "\<S>\<bullet>\<C> \<turnstile> les' : ts _> ts'"
HFail_HInv3: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HFail s s' p"   and "HInv3 s" shows "HInv3 s'"
is_pfa_def: fixes c :: "nat"   and l :: "'b list" shows "is_pfa c l \<equiv> \<lambda>(a, n). \<exists>\<^sub>Al'. a \<mapsto>\<^sub>a l' * \<up> (c = length l' \<and> n \<le> c \<and> l = take n l')"
lm039: fixes a :: "('c \<times> 'd) set" assumes "a \<in> injectionsUniverse" shows "a \<in> injections (Domain a) (Range a)"
Lyndon_fac_set: fixes z :: "'a list"   and w :: "'a list" assumes "z \<in> set (LynFac w)" shows "Lyndon z"
is_natural_1: fixes f :: "'b \<times> 'b" assumes "B.VV.arr f" shows "cmp f \<cdot>\<^sub>B cmp (B.VV.dom f) = cmp f"
grun_elim: fixes A :: "('c, 'd) nba"   and w :: "'c stream"   and V :: "(nat \<times> 'd) set"   and s :: "(nat \<times> 'd) stream"   and v :: "nat \<times> 'd" assumes "grun A w V s v" obtains r :: "'d stream" and k :: "nat" and p :: "'d" where "s = fromN (Suc k) ||| r" and "v = (k, p)"
aux: shows "x ; 1 \<cdot> - (y\<^sup>T ; 1) = x ; 1"
intro: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and a :: "'d" assumes "bidel_arc G M a"   and "bidel_arc_diff_face_axioms M a" shows "bidel_arc_diff_face G M a"
lex_ord_pair_EqD: fixes xs :: "('a \<times> 'd) list"   and ys :: "('a \<times> 'e) list"   and f :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> order option"   and k :: "'a" assumes "oalist_inv_raw xs"   and "oalist_inv_raw ys"   and "lex_ord_pair f xs ys = Some Eq"   and "k \<in> fst ` set xs \<union> fst ` set ys" shows "f k (lookup_pair xs k) (lookup_pair ys k) = Some Eq"
left_residual: fixes x :: "'a"   and a :: "'a"   and b :: "'a" shows "(x * a \<le> b) = (x \<le> a l\<rightarrow> b)"
exhaust: fixes y :: "'a code_status" obtains x1 :: "'a" where "y = CFAILED x1"   | "y = CSUCCESS"   | "y = CFOUND"
magQ_zero: shows "\<lbrakk>0\<rbrakk>\<^sub>Q = (0::'e)"
Seq: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and s :: "'e"   and n :: "nat"   and s' :: "('e, 'f) xstate"   and c\<^sub>2 :: "('e, 'd, 'f) com"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal s\<rangle> =n\<Rightarrow> s'"   and "\<Gamma>\<turnstile> \<langle>c\<^sub>2,s'\<rangle> =n\<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>Seq c\<^sub>1 c\<^sub>2,Normal s\<rangle> =n\<Rightarrow> t"
convol_def: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f" shows "\<langle>f, g\<rangle> \<equiv> \<lambda>a. (f a, g a)"
subd_power_sum: fixes p :: "nat"   and f :: "'c \<Rightarrow> ((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int"   and I :: "'c set" shows "subd p (sum f I) = sum (subd p \<circ> f) I"
map_nth_eq_self: fixes fas :: "'a list"   and l :: "nat" assumes "length fas = l" shows "map ((!) fas) [0..<l] = fas"
no_calls: fixes pn :: "'p" shows "\<forall>pn'. call(pn') \<notin> stermsl (\<Gamma> pn)"
axioms_1: fixes K :: "'a set"   and V :: "'a set"   and R :: "('a, 'b) ring_scheme" assumes "subalgebra K V R" shows "subgroup V (add_monoid R)"
lowner_lub_is_limit: shows "limit_mat f lowner_lub dim"
mult: fixes f1 :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g1 :: "'b \<Rightarrow> 'c"   and f2 :: "'b \<Rightarrow> 'c"   and g2 :: "'b \<Rightarrow> 'c" assumes "f1 \<in> \<Omega>[F](g1)"   and "f2 \<in> \<Omega>[F](g2)" shows "(\<lambda>x. f1 x * f2 x) \<in> \<Omega>[F](\<lambda>x. g1 x * g2 x)"
simps_1: fixes uu :: "value" shows "enumerate_aexp_inputs (L uu) = {}"
tries_of_list_def: fixes key :: "'d \<Rightarrow> 'c list"   and vs :: "'d list" shows "tries_of_list key vs = inserts_tries key vs (Trie None [])"
nat_pow_closed: fixes x :: "nat \<Rightarrow> nat \<Rightarrow> int"   and n :: "nat" assumes "x \<in> carrier (UP Zp)" shows "x [^]\<^bsub>UP Zp\<^esub> n \<in> carrier (UP Zp)"
small_trans: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> l F g"   and "g \<in> l F h" shows "f \<in> l F h"
nat_le: fixes a :: "nat" shows "a - (a + 1) \<le> 0"
triangle_in_hom_2: shows "\<guillemotleft>(g \<star> \<xi> \<cdot> (f \<star> local.inv \<eta> \<star> g) \<cdot> (local.inv \<xi> \<star> f \<star> g)) \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g) : trg g \<star> g \<Rightarrow> g \<star> src g\<guillemotright>"
subgame_maximal_path: fixes V' :: "'a set"   and P :: "'a llist" assumes "V' \<subseteq> V"   and "maximal_path P"   and "lset P \<subseteq> V'" shows "Digraph.maximal_path (subgame V') P"
PiE: fixes f :: "'g \<Rightarrow> 'h"   and A :: "'g set"   and B :: "'g \<Rightarrow> 'h set"   and x :: "'g" assumes "f \<in> Pi A B" obtains "f x \<in> B x"   | "x \<notin> A"
semilattice_order_axioms: shows "semilattice_order (\<sqinter>) (\<le>) (<)"
det_rows_mult: fixes M :: "('c, 'd) sq_matrix"   and T :: "'d set"   and c :: "'d \<Rightarrow> 'c"   and a :: "'d \<Rightarrow> ('c, 'd) vec" shows "Square_Matrix.det (upd_rows M T (\<lambda>i. c i *s a i)) = prod c T * Square_Matrix.det (upd_rows M T a)"
ind: fixes sa :: "nat"   and lm' :: "nat list" assumes "abc_steps_l (0, lm) B n = (sa, lm')" shows "abc_steps_l (length A, lm) (A [+] B) n = (sa + length A, lm')"
rbt_sorted_entries: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "('c, 'd) rbt" assumes "class.linorder less_eq less"   and "ord.rbt_sorted less t" shows "linorder.sorted less_eq (list.map fst (RBT_Impl.entries t))"
rewriteL_comp_comp: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c"   and l :: "'a \<Rightarrow> 'b"   and h :: "'d \<Rightarrow> 'a" assumes "f \<circ> g = l" shows "f \<circ> (g \<circ> h) = l \<circ> h"
len_gt_0: fixes len_of :: "'k itself \<Rightarrow> nat" assumes "class.len len_of" shows "0 < len_of TYPE('k)"
nxt_ev: fixes n :: "nat"   and \<phi> :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream" assumes "(nxt ^^ n) \<phi> xs" shows "ev \<phi> xs"
has_uniq_bottom: fixes P :: "'b set" assumes "codomain.has_bottom P" shows "\<exists>!z. z \<in> P \<and> (\<forall>x\<in>P. codomain.greater_eq x z)"
d0_def: shows "d.d0 = prod_list d.dims0"
prFreshAbs_def: fixes gA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> ('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> 'c"   and SEM :: "('d, 'e, 'f, 'sort, 'g, 'b, 'h) semDom_scheme" shows "prFreshAbs gA SEM \<equiv> \<forall>ys y us s A val val'. wlsAbs (us, s) A \<and> freshAbs ys y A \<and> sWlsVal SEM val \<and> sWlsVal SEM val' \<and> eqBut val val' ys y \<longrightarrow> gA A val = gA A val'"
is_selector_def: fixes sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set" shows "is_selector sel = (\<forall>A B. A \<subseteq> B \<longrightarrow> A \<subseteq> sel A B \<and> sel A B \<subseteq> B)"
step: fixes \<phi> :: "'d stream \<Rightarrow> bool"   and xs :: "'d stream" assumes "ev \<phi> (stl xs)" shows "ev \<phi> xs"
iIN_div_eq1: fixes k :: "nat"   and n :: "nat"   and d :: "nat" assumes "0 < k"   and "n mod k + d mod k < k" shows "[n\<dots>,d] \<oslash> k = [n div k\<dots>,d div k]"
nretkn: shows "\<pi> (k + n) \<noteq> local.return"
execn_sequence_app: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and xs :: "('e, 'd, 'f) com list"   and s :: "'e"   and n :: "nat"   and s' :: "('e, 'f) xstate"   and ys :: "('e, 'd, 'f) com list"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>sequence Seq xs,Normal s\<rangle> =n\<Rightarrow> s'"   and "\<Gamma>\<turnstile> \<langle>sequence Seq ys,s'\<rangle> =n\<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>sequence Seq (xs @ ys),Normal s\<rangle> =n\<Rightarrow> t"
C\<^sub>4_def: shows "C\<^sub>4 = Max (f ` {..n\<^sub>0'})"
mat_eqI: fixes nr :: "nat"   and nc :: "nat"   and m1 :: "'a list list"   and m2 :: "'a list list" assumes "mat nr nc m1"   and "mat nr nc m2"   and "\<And>i j. \<lbrakk>i < nc; j < nr\<rbrakk> \<Longrightarrow> m1 ! i ! j = m2 ! i ! j" shows "m1 = m2"
Abs_unit_inject: fixes x :: "bool"   and y :: "bool" assumes "x \<in> {True}"   and "y \<in> {True}" shows "(Abs_unit x = Abs_unit y) = (x = y)"
rhs: shows "V_axiom_on A f \<and> preorder_on A (rwp_on A f) \<and> total_on A (rwp_on A f)"
green_typeI_typeII_chain_def: fixes s :: "(real \<times> real) set"   and i :: "real \<times> real"   and j :: "real \<times> real"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and two_chain_typeI :: "(real \<times> real \<Rightarrow> real \<times> real) set"   and two_chain_typeII :: "(real \<times> real \<Rightarrow> real \<times> real) set" shows "green_typeI_typeII_chain s i j F two_chain_typeI two_chain_typeII \<equiv> R2 i j \<and> green_typeI_chain i j F two_chain_typeI s \<and> green_typeII_chain i j F two_chain_typeII s"
continuous_on_closed_gen: fixes f :: "'b \<Rightarrow> 'a"   and S :: "'b set"   and T :: "'a set" assumes "f ` S \<subseteq> T" shows "continuous_on S f = (\<forall>U. closedin (top_of_set T) U \<longrightarrow> closedin (top_of_set S) (S \<inter> f -` U))"
hyps_3: shows "\<forall>s'. \<Gamma>\<turnstile> \<langle>c1,Normal s\<rangle> \<Rightarrow> Abrupt s' \<longrightarrow> \<Gamma>\<turnstile>c2 \<down> Normal s' \<and> \<Gamma>\<turnstile>mark_guards f c2 \<down> Normal s'"
simps_10: fixes v :: "Formula.formula" shows "remove_neg (formula.Exists v) = formula.Exists v"
src_trg: fixes \<mu> :: "('a S.E.term, 'a) S.arr" assumes "S.arr \<mu>" shows "S.src (S.trg \<mu>) = S.trg \<mu>"