hcju/isabelleps huggingface dataset:
parallel_fixp_induct_1_1: fixes orda :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and luba :: "'h set \<Rightarrow> 'h"   and ordb :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and lubb :: "'i set \<Rightarrow> 'i"   and F :: "('j \<Rightarrow> 'h) \<Rightarrow> 'j \<Rightarrow> 'h"   and G :: "('k \<Rightarrow> 'i) \<Rightarrow> 'k \<Rightarrow> 'i"   and f :: "'j \<Rightarrow> 'h"   and g :: "'k \<Rightarrow> 'i"   and P :: "('j \<Rightarrow> 'h) \<Rightarrow> ('k \<Rightarrow> 'i) \<Rightarrow> bool" assumes "partial_function_definitions orda luba"   and "partial_function_definitions ordb lubb"   and "\<And>x. monotone (fun_ord orda) orda (\<lambda>f. F f x)"   and "\<And>y. monotone (fun_ord ordb) ordb (\<lambda>g. G g y)"   and "f \<equiv> ccpo.fixp (fun_lub luba) (fun_ord orda) F"   and "g \<equiv> ccpo.fixp (fun_lub lubb) (fun_ord ordb) G"   and "ccpo.admissible (prod_lub (fun_lub luba) (fun_lub lubb)) (rel_prod (fun_ord orda) (fun_ord ordb)) (\<lambda>x. P (fst x) (snd x))"   and "P (\<lambda>_. luba {}) (\<lambda>_. lubb {})"   and "\<And>f g. P f g \<Longrightarrow> P (F f) (G g)" shows "P f g"
QI_w: fixes i :: "int"   and i' :: "int"   and j :: "int"   and j' :: "int" assumes "i \<le> i'"   and "i' < j"   and "j \<le> j'" shows "w i j + w i' j' \<le> w i' j + w i j'"
main_part: fixes S :: "'a set"   and M :: "'a measure" assumes "S \<in> sets M" shows "main_part M S = S"
comp_in_homI': fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and a :: "'b"   and c :: "'b" assumes "category C"   and "partial_magma.arr C f"   and "partial_magma.arr C g"   and "partial_magma.dom C f = a"   and "partial_magma.cod C g = c"   and "partial_magma.dom C g = partial_magma.cod C f" shows "partial_magma.in_hom C (C g f) a c"
intros: fixes \<sigma> :: "'b"   and W :: "'b set"   and \<Sigma>' :: "'b set"   and \<Sigma> :: "'b set"   and R :: "('b \<times> 'b) set"   and W' :: "'b set" assumes "\<sigma> \<in> W"   and "\<Sigma>' = \<Sigma> \<union> R `` {\<sigma>}"   and "W' = W - {\<sigma>} \<union> (R `` {\<sigma>} - \<Sigma>)" shows "((\<Sigma>, W), \<Sigma>', W') \<in> sse_step R"
set_iterator_genord_sng_correct: fixes x :: "'e"   and R :: "'e \<Rightarrow> 'e \<Rightarrow> bool" shows "set_iterator_genord (set_iterator_sng x) {x} R"
rep_eq: fixes x :: "'b segment"   and xa :: "'b segment" shows "Rep_segment (x \<oplus> xa) = (if snd (Rep_segment x) = fst (Rep_segment xa) then (fst (Rep_segment x), snd (Rep_segment xa)) else Rep_segment x)"
LNil_eq_lappend_iff: fixes xs :: "'a llist"   and ys :: "'a llist" shows "([]\<^sub>l = xs @\<^sub>l ys) = (xs = []\<^sub>l \<and> ys = []\<^sub>l)"
t: shows "\<forall>T. gap i T \<le> gap i (LEAST t. \<forall>T. gap i T \<le> gap i t)"
select_convs_3: fixes Functions :: "'c \<Rightarrow> (real, 'b) vec \<Rightarrow> real"   and Predicates :: "'b \<Rightarrow> (real, 'b) vec \<Rightarrow> bool"   and Contexts :: "'a \<Rightarrow> ((real, 'b) vec \<times> (real, 'b) vec) set \<Rightarrow> ((real, 'b) vec \<times> (real, 'b) vec) set"   and Programs :: "'b \<Rightarrow> (((real, 'b) vec \<times> (real, 'b) vec) \<times> (real, 'b) vec \<times> (real, 'b) vec) set"   and ODEs :: "'b \<Rightarrow> (real, 'b) vec \<Rightarrow> (real, 'b) vec"   and ODEBV :: "'b \<Rightarrow> 'b set"   and more :: "'d" shows "Contexts \<lparr>Functions = Functions, Predicates = Predicates, Contexts = Contexts, Programs = Programs, ODEs = ODEs, ODEBV = ODEBV, \<dots> = more\<rparr> = Contexts"
support_on_simps_1: fixes f :: "'a \<Rightarrow> 'b" shows "support_on {} f = {}"
tendsto_uminus_ereal: fixes f :: "'a \<Rightarrow> ereal"   and x :: "ereal"   and F :: "'a filter" assumes "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. - f x) \<longlongrightarrow> - x) F"
create_name_gba_aimpl_def: fixes \<phi> :: "'b ltlr" shows "create_name_gba_aimpl \<phi> \<equiv> create_graph_aimpl \<phi> \<bind> (\<lambda>nds. ASSERT (nds_invars nds) \<bind> (\<lambda>_. cr_rename_gba nds \<phi>))"
Soundness: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and Doc :: "character list"   and \<alpha> :: "symbol list" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel"   and "LocalLexing.finished_item \<SS> Doc \<alpha> \<in> LocalLexing.\<II> \<TT> \<RR> \<SS> Lex Sel Doc" shows "\<exists>p. LocalLexing.pvalid_with \<NN> \<TT> \<RR> \<SS> Doc p (LocalLexing.finished_item \<SS> Doc \<alpha>) 0 [] \<and> p \<in> LocalLexing.ll \<NN> \<TT> \<RR> \<SS> Lex Sel Doc"
onll_ostep_invariantI: fixes A :: "('a \<times> ('b, 'c, 'd, 'e) seqp, 'f) automaton"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> 'f \<Rightarrow> bool"   and U :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<Gamma> :: "'d \<Rightarrow> ('b, 'c, 'd, 'e) seqp"   and P :: "('a \<times> 'e) \<times> 'f \<times> 'a \<times> 'e \<Rightarrow> bool" assumes "\<And>\<sigma> p l a \<sigma>' p' l'. \<lbrakk>(\<sigma>, p) \<in> oreachable A S U; ((\<sigma>, p), a, \<sigma>', p') \<in> automaton.trans A; S \<sigma> \<sigma>' a; l \<in> labels \<Gamma> p; l' \<in> labels \<Gamma> p'\<rbrakk> \<Longrightarrow> P ((\<sigma>, l), a, \<sigma>', l')" shows "A \<Turnstile>\<^sub>A (S, U \<rightarrow>) onll \<Gamma> P"
insert_if: fixes A :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "finite A" shows "prod g (insert x A) = (if x \<in> A then prod g A else g x * prod g A)"
sp_subset_def: fixes s :: "'a state_t_scheme" shows "sp_subset s \<equiv> (\<forall>p1 p2. sp_impl_subj_subj s p1 p2 \<longrightarrow> Policy.sp_spec_subj_subj p1 p2) \<and> (\<forall>p1 p2 m. sp_impl_subj_obj s p1 p2 m \<longrightarrow> Policy.sp_spec_subj_obj p1 p2 m)"
cong_scalar_left: fixes b :: "'b"   and c :: "'b"   and a :: "'b"   and d :: "'b" assumes "[b = c] (mod a)" shows "[d * b = d * c] (mod a)"
simps_4: fixes a\<^sub>1 :: "aexp"   and a\<^sub>2 :: "aexp"   and s :: "char list \<Rightarrow> int option" shows "paval' (Times a\<^sub>1 a\<^sub>2) s = opttimes (paval' a\<^sub>1 s) (paval' a\<^sub>2 s)"
ball_eq_greaterThanLessThan: fixes a :: "real"   and b :: "real" shows "ball a b = {a - b<..<a + b}"
substitutes_on_unilateral_substitutes_on: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "substitutes_on A f" shows "unilateral_substitutes_on A f"
eWlsBinp_simp1: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> 'd option" shows "eWlsBinp MOD delta (OKI binp) = igWlsBinp MOD delta binp"
simps_6: fixes f1 :: "String.literal \<Rightarrow> typ list \<Rightarrow> 'a"   and f2 :: "variable \<Rightarrow> String.literal set \<Rightarrow> 'a"   and x21 :: "variable"   and x22 :: "String.literal set" shows "(case Tv x21 x22 of Ty x xa \<Rightarrow> f1 x xa | Tv x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
simps_2: fixes e :: "char list exp" shows "blocks ([], [], [], e) = e"
hull_eq: fixes S :: "'c set \<Rightarrow> bool"   and s :: "'c set" assumes "\<And>T. Ball T S \<Longrightarrow> S (\<Inter> T)" shows "(S hull s = s) = S s"
index_less_size_conv: fixes xs :: "'b list"   and x :: "'b" shows "(index xs x < length xs) = (x \<in> set xs)"
pow_ge_zero: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and n :: "nat" assumes "SN_Orders.class.ordered_semiring_1 one times plus zero less_eq"   and "less_eq zero a" shows "less_eq zero (power.power one times a n)"
new_character_data_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_character_data_ptr :: "(_) character_data_ptr"   and ptr :: "(_) element_ptr"   and getter :: "('z, 'x, 'y, 'aa, 'ad option) RElement_scheme \<Rightarrow> 'ah" assumes "h \<turnstile> new_character_data \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_character_data \<rightarrow>\<^sub>r new_character_data_ptr" shows "preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h h'"
sys_xcpts_def: shows "sys_xcpts \<equiv> {NullPointer, ClassCast, OutOfMemory}"
f_def: shows "f \<equiv> \<lambda>(k, q). the (fst ((p ## r) !! k) q)"
the_lub_def: fixes A :: "'b set" shows "\<Squnion>A = The (lub A)"
weakStepSimAux_def: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi" shows "weakStepSimAux P Rel Q \<equiv> (\<forall>Q' a x. Q \<longmapsto> a<\<nu>x> \<prec> Q' \<and> x \<sharp> P \<longrightarrow> (\<exists>P'. P \<Longrightarrow>\<^sub>l a<\<nu>x> \<prec> P' \<and> (P', Q') \<in> Rel)) \<and> (\<forall>Q' a x. Q \<longmapsto> a<x> \<prec> Q' \<and> x \<sharp> P \<longrightarrow> (\<exists>P''. \<forall>u. \<exists>P'. P \<Longrightarrow>\<^sub>lu in P''\<rightarrow>a<x> \<prec> P' \<and> (P', Q'[x::=u]) \<in> Rel)) \<and> (\<forall>Q' \<alpha>. Q \<longmapsto> \<alpha> \<prec> Q' \<longrightarrow> (\<exists>P'. P \<Longrightarrow>\<^sub>l \<alpha> \<prec> P' \<and> (P', Q') \<in> Rel))"
finite_UNIV_lit_def: shows "finite_UNIV = Phantom(lit) False"
distinct_p_divisors: fixes P :: "('a \<Rightarrow> ant) set"   and P' :: "('a \<Rightarrow> ant) set" assumes "P \<in> Pds"   and "P' \<in> Pds" shows "(P \<noteq> P') = (\<not> v_equiv K (\<nu>\<^bsub>K P\<^esub>) (\<nu>\<^bsub>K P'\<^esub>))"
naturality: fixes f :: "'a \<times> 'a \<times> 'a" assumes "CCC.arr f" shows "\<alpha> (CCC.cod f) \<cdot> T.ToTC f = T.ToCT f \<cdot> \<alpha> (CCC.dom f)"
star_n_inverse: fixes X :: "nat \<Rightarrow> 'b" shows "inverse (star_n X) = star_n (\<lambda>n. inverse (X n))"
AssumeH_8: fixes A1 :: "fm"   and A2 :: "fm"   and A3 :: "fm"   and A4 :: "fm"   and A5 :: "fm"   and A6 :: "fm"   and A7 :: "fm"   and A8 :: "fm"   and H :: "fm set" shows "insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 H))))))) \<turnstile> A8"
wt_elim_cases_6: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and T' :: "ty"   and e :: "expr"   and T :: "tys" assumes "E,dt\<Turnstile>In1l (Cast T' e)\<Colon>T" obtains Ta :: "ty" where "T = Inl T'" and "E,dt\<Turnstile>e\<Colon>-Ta" and "is_acc_type (prg E) (pkg E) T'" and "prg E\<turnstile>Ta\<preceq>? T'"
Datisi: fixes A :: "'a set"   and C :: "'a set"   and B :: "'a set" assumes "A Q C"   and "B I C" shows "A I B"
Rep_bd_type_F_inverse: fixes x :: "bd_type_F" shows "Abs_bd_type_F (Rep_bd_type_F x) = x"
set_to_map_simp: fixes S :: "('g \<times> 'h) set"   and k :: "'g"   and v :: "'h" assumes "inj_on fst S" shows "(set_to_map S k = Some v) = ((k, v) \<in> S)"
spref_lq: fixes u :: "'b list"   and v :: "'b list"   and w :: "'b list" assumes "u \<le>p v"   and "v <p w" shows "u\<inverse>\<^sup>>v <p u\<inverse>\<^sup>>w"
b': shows "emb b B \<in> A \<union> B \<rightarrow> carrier K"
is_red_addsI: fixes f :: "'t \<Rightarrow>\<^sub>0 'c"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and v :: "'t"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "f \<in> F"   and "f \<noteq> 0"   and "v \<in> keys p"   and "lt f adds\<^sub>t v" shows "is_red F p"
sg_mult_closed: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and x :: "'e"   and y :: "'e" assumes "Group G"   and "G \<guillemotright> H "   and "x \<in> H"   and "y \<in> H" shows "x \<cdot>\<^bsub>G\<^esub> y \<in> H"
cinner_real: fixes x :: "'a" shows "x \<bullet>\<^sub>C x \<in> \<real>"
prems_2: shows "subst_domain (\<sigma>1(v := t)) \<inter> subst_domain \<theta>2 = {}"
monoid_homomorphism_\<eta>_cong: fixes \<eta>' :: "'a \<Rightarrow> 'a set" assumes "\<And>x. x \<in> G \<Longrightarrow> \<eta>' x = kernel.Class x"   and "\<eta>' \<in> extensional G" shows "monoid_homomorphism \<eta>' G (\<cdot>) \<one> kernel.Partition kernel.quotient_composition (kernel.Class \<one>)"
assms_3: assumes "dip \<in> kD (rt \<xi>')"   and "rt \<xi> \<sqsubseteq>\<^bsub>dip\<^esub> rt \<xi>'"   and "sqn (rt \<xi>) dip \<le> sqn (rt \<xi>') dip" shows "R dip \<xi> \<xi>'"
conductor_ge_iff: fixes a :: "nat" shows "(a \<le> conductor) = (\<forall>d. induced_modulus d \<longrightarrow> a \<le> d)"
cpI3: fixes f :: "('a state \<times> 'a state \<Rightarrow> 'b) \<Rightarrow> ('a state \<times> 'a state \<Rightarrow> 'c) \<Rightarrow> ('a state \<times> 'a state \<Rightarrow> 'd) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> 'e"   and P :: "('a state \<times> 'a state \<Rightarrow> 'f) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> 'b"   and Q :: "('a state \<times> 'a state \<Rightarrow> 'f) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> 'c"   and R :: "('a state \<times> 'a state \<Rightarrow> 'f) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> 'd" assumes "\<forall>X Y Z \<tau>. f X Y Z \<tau> = f (\<lambda>_. X \<tau>) (\<lambda>_. Y \<tau>) (\<lambda>_. Z \<tau>) \<tau>"   and "cp P"   and "cp Q"   and "cp R" shows "cp (\<lambda>X. f (P X) (Q X) (R X))"
induct: fixes P :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a1 :: "'a list" assumes "\<And>f. P f []"   and "\<And>f x. P f [x]"   and "\<And>f v vb vc. \<lbrakk>P f (take (length (v # vb # vc) div 2) (v # vb # vc)); P f (drop (length (v # vb # vc) div 2) (v # vb # vc))\<rbrakk> \<Longrightarrow> P f (v # vb # vc)" shows "P a0 a1"
remdups_adj_append: fixes xs\<^sub>1 :: "'c list"   and x :: "'c"   and xs\<^sub>2 :: "'c list" shows "remdups_adj (xs\<^sub>1 @ x # xs\<^sub>2) = remdups_adj (xs\<^sub>1 @ [x]) @ tl (remdups_adj (x # xs\<^sub>2))"
valid_stepI: fixes r :: "'r"   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"   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" assumes "revision_step r s s'" shows "s \<leadsto> s'"
fps_divide_uminus: fixes f :: "'a fps"   and g :: "'a fps" shows "- f / g = - (f / g)"
Image_empty1: fixes X :: "'f set" shows "{} `` X = {}"
\<R>_sym: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and bexp_neg :: "'d \<Rightarrow> 'd"   and assign_post :: "'d set \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'd set"   and dma_type :: "'a \<Rightarrow> 'd set"   and pred_False :: "'d"   and \<Gamma> :: "'a \<Rightarrow> 'd set option"   and \<S> :: "'a set \<times> 'a set"   and P :: "'d set" assumes "sifum_types ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type pred_False" shows "sym (sifum_types.\<R> ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type \<Gamma> \<S> P)"
stoch_proc_filt_disc_filtr: fixes X :: "nat \<Rightarrow> 'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure" assumes "\<And>i. X i \<in> M \<rightarrow>\<^sub>M N" shows "disc_filtr M (stoch_proc_filt M X N)"
refl_O_iff: fixes A :: "('b \<times> 'b) set" shows "refl A = (Id \<subseteq> A)"
lconf_upd2: 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 h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "'a \<Rightarrow> val option"   and E :: "'a \<Rightarrow> ty option"   and v :: "val"   and T :: "ty"   and V :: "'a" assumes "P,h \<turnstile> l (:\<le>)\<^sub>w E"   and "P,h \<turnstile> v :\<le> T" shows "P,h \<turnstile> l(V \<mapsto> v) (:\<le>)\<^sub>w E(V \<mapsto> T)"
assms: fixes S :: "'a set" assumes "S \<in> \<F>" shows "openin (top_of_set U) S"
block_design_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int" shows "block_design point_set block_collection u_block_size \<equiv> proper_design point_set block_collection \<and> block_design_axioms block_collection u_block_size"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>D) (\<cdot>\<^sub>D) GF.map D.map \<eta>'.map"
orthogonal_complement_def: fixes n :: "nat"   and W :: "'a vec set" shows "vec_module.orthogonal_complement n W = {x \<in> carrier_vec n. \<forall>y\<in>W. x \<bullet> y = (0::'a)}"
first_countableI: fixes "open" :: "'b set \<Rightarrow> bool"   and \<A> :: "'b set set"   and x :: "'b" assumes "class.topological_space open"   and "countable \<A>"   and "\<And>A. A \<in> \<A> \<Longrightarrow> x \<in> A"   and "\<And>A. A \<in> \<A> \<Longrightarrow> open A"   and "\<And>S. \<lbrakk>open S; x \<in> S\<rbrakk> \<Longrightarrow> \<exists>A\<in>\<A>. A \<subseteq> S" shows "\<exists>\<A>. (\<forall>i. x \<in> \<A> i \<and> open (\<A> i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. \<A> i \<subseteq> S))"
amax_mem_m: fixes x :: "ant"   and y :: "ant" assumes "x \<in> Z\<^sub>-\<^sub>\<infinity>"   and "y \<in> Z\<^sub>-\<^sub>\<infinity>" shows "amax x y \<in> Z\<^sub>-\<^sub>\<infinity>"
connected_def: fixes v :: "'c"   and w :: "'c" shows "T.connected v w \<equiv> \<exists>xs. T.walk xs \<and> xs \<noteq> [] \<and> hd xs = v \<and> last xs = w"
not_loose_bvar_imp_not_loose_bvar1: fixes t :: "Core.term"   and n :: "nat" assumes "\<not> loose_bvar t n" shows "\<not> loose_bvar1 t n"
wf: fixes s :: "'a" assumes "\<And>s'. (s', s) \<in> {(snd (snd g s), s) |s. fst g s} \<Longrightarrow> thesis s'" shows "thesis s"
in_hom_char: fixes f :: "'a"   and a :: "'a"   and b :: "'a" shows "hom'.in_hom f a b = (hom'.arr a \<and> hom'.arr b \<and> hom'.arr f \<and> \<guillemotleft>f : a \<rightarrow> b\<guillemotright>)"
lower_rank_of_def: fixes carrier' :: "'a set" shows "rank_of carrier' \<equiv> Min {card B |B. basis_in carrier' B}"
pos_via_half_dist: fixes a :: "real"   and b :: "real" assumes "dist a b < b / 2"   and "0 < b" shows "0 < a"
smult_1_left: fixes p :: "'d poly" shows "smult (1::'d) p = p"
real_of_float_cases: fixes y :: "real" assumes "y \<in> float" obtains x :: "float" where "y = real_of_float x"
AddSA_CompFun_ran2: fixes SA1 :: "('d, 'e, 'f) seqauto"   and A :: "('d, 'e, 'f) hierauto"   and SA2 :: "('d, 'e, 'f) seqauto"   and S :: "'d"   and T :: "'d" assumes "States SA1 \<inter> HAStates A = {}"   and "States SA2 \<inter> (HAStates A \<union> States SA1) = {}"   and "S \<in> HAStates A"   and "T \<in> States SA1" shows "ran (CompFun (A [++] (S, SA1) [++] (T, SA2))) = insert {} (insert {SA2} (ran (CompFun (A [++] (S, SA1)))))"
F_simp: fixes g :: "'d" assumes "D.arr g" shows "F g = initial_arrow_to_functor.the_ext (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G (Fo (D.dom g)) (\<eta>o (D.dom g)) (Fo (D.cod g)) (\<eta>o (D.cod g) \<cdot>\<^sub>D g)"
invar_tree0: fixes x :: "'b" shows "invar_tree (Node 0 x [])"
simps_1: fixes x :: "brick list" shows "domain_wall (basic x) = domain_block x"
bind_simps_2: fixes x :: "'b"   and xs :: "'b list"   and f :: "'b \<Rightarrow> 'a list" shows "List.bind (x # xs) f = f x @ List.bind xs f"
simps_24: fixes vc :: "bexp"   and vd :: "'a acom"   and ve :: "'a acom"   and vf :: "'a"   and v :: "char list"   and va :: "aexp"   and vb :: "'a" shows "(IF vc THEN vd ELSE ve {vf} \<sqsubseteq> v ::= va {vb}) = False"
cexpr_typing_expr_rf_to_cexpr: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "expr"   and t :: "pdf_type" assumes "\<Gamma> \<turnstile> e : t"   and "randomfree e" shows "\<Gamma> \<turnstile>\<^sub>c expr_rf_to_cexpr e : t"
abs_eq: fixes xa :: "'a"   and x :: "'b" shows "Poly_Mapping.single xa x = Abs_poly_mapping (\<lambda>k'. x when xa = k')"
complete_space_with_middle_def: shows "complete_space_with_middle TYPE('b) \<equiv> \<forall>x y. \<exists>z. dist x z = dist x y / 2 \<and> dist z y = dist x y / 2"
divide_pos_pos: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero x"   and "less zero y" shows "less zero (divide x y)"
cases: fixes x :: "('b \<times> color) tree \<times> 'b \<times> ('b \<times> color) tree" obtains t1 :: "('b \<times> color) tree" and a :: "'b" and t2 :: "('b \<times> color) tree" and b :: "'b" and t3 :: "('b \<times> color) tree" where "x = (t1__, a__, R t2__ b__ t3__)"   | t1 :: "('b \<times> color) tree" and a :: "'b" and t2 :: "('b \<times> color) tree" and b :: "'b" where "x = (B t1a__ aa__ t2a__, ba__, \<langle>\<rangle>)"   | t1 :: "('b \<times> color) tree" and a :: "'b" and t2 :: "('b \<times> color) tree" and b :: "'b" and v :: "('b \<times> color) tree" and vc :: "'b" and vb :: "('b \<times> color) tree" where "x = (B t1b__ ab__ t2b__, bb__, B v__ vc__ vb__)"   | t1 :: "('b \<times> color) tree" and a :: "'b" and t2 :: "('b \<times> color) tree" and b :: "'b" and t3 :: "('b \<times> color) tree" and c :: "'b" where "x = (R t1c__ ac__ (B t2c__ bc__ t3a__), c__, \<langle>\<rangle>)"   | t1 :: "('b \<times> color) tree" and a :: "'b" and t2 :: "('b \<times> color) tree" and b :: "'b" and t3 :: "('b \<times> color) tree" and c :: "'b" and v :: "('b \<times> color) tree" and vc :: "'b" and vb :: "('b \<times> color) tree" where "x = (R t1 ad__ (B t2 b t3), c, B va__ vca__ vba__)"   | a :: "'b" where "x = (\<langle>\<rangle>, ae__, \<langle>\<rangle>)"   | v :: "('b \<times> color) tree" and vc :: "'b" and a :: "'b" where "x = (R vd__ vcb__ \<langle>\<rangle>, af__, \<langle>\<rangle>)"   | v :: "('b \<times> color) tree" and vc :: "'b" and va :: "('b \<times> color) tree" and vf :: "'b" and ve :: "('b \<times> color) tree" and a :: "'b" where "x = (R ve__ vcc__ (R vaa__ vf__ ve), ag__, \<langle>\<rangle>)"   | a :: "'b" and v :: "('b \<times> color) tree" and vc :: "'b" and vb :: "('b \<times> color) tree" where "x = (\<langle>\<rangle>, ah__, B vg__ vcd__ vbb__)"   | va :: "('b \<times> color) tree" and vf :: "'b" and a :: "'b" and v :: "('b \<times> color) tree" and vc :: "'b" and vb :: "('b \<times> color) tree" where "x = (R vab__ vfa__ \<langle>\<rangle>, ai__, B vh__ vce__ vbc__)"   | va :: "('b \<times> color) tree" and vf :: "'b" and vd :: "('b \<times> color) tree" and vi :: "'b" and vh :: "('b \<times> color) tree" and a :: "'b" and v :: "('b \<times> color) tree" and vc :: "'b" and vb :: "('b \<times> color) tree" where "x = (R va vf (R vd vi vh), a, B v vc vb)"
upto_strict_1: shows "supto\<cdot>\<bottom> = \<bottom>"
Fin: assumes "finite A"   and "|A| <o |UNIV|" shows "thesis"
cf_pos_poly_def: fixes f :: "int poly" shows "cf_pos_poly f = (let c = content f in Let (sgn (lead_coeff f) * c) (sdiv_poly f))"
f21: shows "t' \<otimes> t \<otimes> (snd x \<otimes> snd x') \<otimes> (r \<otimes> r') = t' \<otimes> (t \<otimes> s \<otimes> fst x) \<otimes> snd x' \<otimes> r'"
ccSUP_inf: fixes B :: "'d set"   and f :: "'d \<Rightarrow> 'a"   and a :: "'a" assumes "countable B" shows "inf (Sup (f ` B)) a = (SUP b\<in>B. inf (f b) a)"
multiset_eqI: fixes A :: "'b multiset"   and B :: "'b multiset" assumes "\<And>x. count A x = count B x" shows "A = B"
product_group_def: fixes I :: "'e set"   and G :: "'e \<Rightarrow> ('f, 'g) monoid_scheme" shows "product_group I G \<equiv> \<lparr>carrier = \<Pi>\<^sub>E i\<in>I. carrier (G i), monoid.mult = \<lambda>x y. \<lambda>i\<in>I. x i \<otimes>\<^bsub>G i\<^esub> y i, one = \<lambda>i\<in>I. \<one>\<^bsub>G i\<^esub>\<rparr>"