hcju/isabelleps huggingface dataset:
q_dvd_prod_list_var_prop: fixes q :: "real poly"   and qs :: "real poly list" assumes "q \<in> set qs"   and "q \<noteq> 0" shows "q dvd prod_list_var qs"
remdups_mset_eq_empty: fixes D :: "'a multiset" shows "(remdups_mset D = {#}) = (D = {#})"
Ch_iiaD: fixes a :: "'x"   and x :: "'h"   and B :: "'x set"   and C :: "'x set" assumes "a \<in> Ch x B"   and "B \<subseteq> A"   and "C \<subseteq> B"   and "a \<in> C" shows "a \<in> Ch x C"
wellformed: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and b :: "'b set" assumes "incidence_system point_set block_collection"   and "b \<in># block_collection" shows "b \<subseteq> point_set"
Complex_Re_express_cnj: fixes z :: "complex" shows "Complex (Re z) 0 = (z + cnj z) / 2"
infsetsum'_converges_cofin_subset: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and F :: "'c set" assumes "infsetsum'_converges f A"   and "finite F" shows "infsetsum'_converges f (A - F)"
WOC_wrt_preorder_iff_reduction_correspondence_simulation: fixes TRel :: "('procT \<times> 'procT) set" shows "(weakly_operational_corresponding TRel \<and> preorder TRel \<and> weak_reduction_correspondence_simulation TRel Target) = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> TRel = _Collect (T1, T2) ((TargetTerm T1, TargetTerm T2) \<in> Rel) \<and> (\<forall>S T. (SourceTerm S, TargetTerm T) \<in> Rel \<longrightarrow> (\<lbrakk>S\<rbrakk>, T) \<in> TRel) \<and> preorder Rel \<and> weak_reduction_correspondence_simulation Rel (STCal Source Target))"
comm_monoid_mset_axioms: shows "comm_monoid_mset (+) (0::'a)"
a: shows "[\<^bold>\<A>(F \<^bold>= G) in v]"
co_xs: shows "cyclic_on (perm_restrict f (set xs)) (set xs)"
finite_enumerate_Ex: fixes S :: "'a set"   and s :: "'a" assumes "finite S"   and "s \<in> S" shows "\<exists>n<card S. enum S n = s"
l_get_tag_name_axioms: shows "l_get_tag_name ShadowRootClass.type_wf get_tag_name get_tag_name_locs"
HLD_def: fixes s :: "'c set" shows "HLD s = holds (\<lambda>x. x \<in> s)"
prj0_simps_2: fixes f :: "'a"   and g :: "'a" assumes "cospan f g" shows "local.dom \<p>\<^sub>0[f, g] = f \<down>\<down> g"
fact_div_fact_ffact: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "fact n div fact m = ffact (n - m) n"
nb\<^sub>1: fixes op\<^sub>1' :: "('variable \<times> 'domain) strips_operator"   and op\<^sub>2' :: "('variable \<times> 'domain) strips_operator" assumes "op\<^sub>1' \<in> set ops'"   and "op\<^sub>2' \<in> set ops'" shows "STRIPS_Semantics.are_operator_effects_consistent op\<^sub>1' op\<^sub>2'"
sorted_wrt_iff_nth_Suc_transp: fixes P :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and xs :: "'g list" assumes "transp P" shows "sorted_wrt P xs = (\<forall>i. Suc i < length xs \<longrightarrow> P (xs ! i) (xs ! Suc i))"
assms: shows "infer \<Gamma> (Snd P) = Some \<tau>"
of_real_eq_1_iff: fixes x :: "real" shows "(of_real x = (1::'a)) = (x = 1)"
base: fixes s :: "'a"   and S :: "'a set" assumes "s \<in> insert (0::'a) S" shows "s \<in> group_closure S"
induct: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x1 :: "'a"   and x2 :: "'a"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "r\<^sup>*\<^sup>* x1 x2"   and "\<And>a. P a a"   and "\<And>a b c. \<lbrakk>r\<^sup>*\<^sup>* a b; P a b; r b c\<rbrakk> \<Longrightarrow> P a c" shows "P x1 x2"
eq: fixes n :: "nat" shows "(LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f x) - (LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * g x) = LINT x|lebesgue_on {- pi..pi}. sin ((real n + 1 / 2) * x) * ((if \<bar>x\<bar> < d then 0 else f x - g x) / sin (x / 2) / 2)"
meld_correct_2: fixes q :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewBinomialTree list"   and q' :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewBinomialTree list" assumes "SkewBinomialHeapStruc.invar (map BsSkewBinomialHeapStruc.bsmapt q)"   and "SkewBinomialHeapStruc.invar (map BsSkewBinomialHeapStruc.bsmapt q')" shows "image_mset fst (SkewBinomialHeapStruc.queue_to_multiset (map BsSkewBinomialHeapStruc.bsmapt (BsSkewBinomialHeapStruc.meld q q'))) = image_mset fst (SkewBinomialHeapStruc.queue_to_multiset (map BsSkewBinomialHeapStruc.bsmapt q)) + image_mset fst (SkewBinomialHeapStruc.queue_to_multiset (map BsSkewBinomialHeapStruc.bsmapt q'))"
lookup_pair_eq_valueI: fixes xs :: "('a \<times> 'c) list"   and k :: "'a"   and v :: "'c" assumes "oalist_inv_raw xs"   and "(k, v) \<in> set xs" shows "lookup_pair xs k = v"
set_rcs_def: fixes G :: "('c, 'd) Group_scheme"   and H :: "'c set" shows "set_rcs G H = {C. \<exists>a\<in>carrier G. C = H \<bullet>\<^bsub>G\<^esub> a}"
preserves_dom: fixes f :: "'a" assumes "hom'.arr f" shows "hom'.dom (\<psi>.map f) = FG.map (hom'.dom f)"
IH: assumes "insert_ops (pre @ suf)" shows "insert_ops suf"
interior_of_unique: fixes T :: "'d set"   and S :: "'d set"   and X :: "'d topology" assumes "T \<subseteq> S"   and "openin X T"   and "\<And>T'. \<lbrakk>T' \<subseteq> S; openin X T'\<rbrakk> \<Longrightarrow> T' \<subseteq> T" shows "X interior_of S = T"
case_eq_if: fixes f :: "'b \<Rightarrow> nat \<Rightarrow> 'a"   and tm :: "'b tm" shows "(case tm of ??.Time_Monad.tm.TM x xa \<Rightarrow> f x xa) = f (tm.val tm) (un_TM2 tm)"
enabled_force_enabled: fixes x :: "'s"   and \<omega> :: "'s stream" shows "enabled x (force_enabled x \<omega>)"
eo: shows "prod_list (map ((!) sign) xa) = 1 \<or> prod_list (map ((!) sign) xa) = 0 \<or> prod_list (map ((!) sign) xa) = - 1"
finprod_reindex: fixes f :: "'e \<Rightarrow> 'a"   and h :: "'f \<Rightarrow> 'e"   and A :: "'f set" assumes "f \<in> h ` A \<rightarrow> carrier G"   and "inj_on h A" shows "finprod G f (h ` A) = (\<Otimes>x\<in>A. f (h x))"
cInf_lessD: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<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 X :: "'b set"   and z :: "'b" assumes "class.conditionally_complete_linorder Inf Sup inf less_eq less sup"   and "X \<noteq> {}"   and "less (Inf X) z" shows "\<exists>x\<in>X. less x z"
abs_eq: fixes x :: "\<nu> option" shows "rep (make\<kappa> x) = the x"
partitions_def: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat" shows "(p partitions n) = ((\<forall>i. p i \<noteq> 0 \<longrightarrow> 1 \<le> i \<and> i \<le> n) \<and> (\<Sum>i\<le>n. p i * i) = n)"
pairwise_rel_prime2: shows "\<forall>q. List.member qs2 q \<longrightarrow> coprime p q"
length_listsetD: fixes vs :: "'a list"   and VS :: "'a set list" assumes "vs \<in> listset VS" shows "length vs = length VS"
banach_fix: fixes s :: "'a set"   and c :: "real"   and f :: "'a \<Rightarrow> 'a" assumes "complete s"   and "s \<noteq> {}"   and "0 \<le> c"   and "c < 1"   and "f ` s \<subseteq> s"   and "\<forall>x\<in>s. \<forall>y\<in>s. dist (f x) (f y) \<le> c * dist x y" shows "\<exists>!x. x \<in> s \<and> f x = x"
4: shows "DOM f = Cod (Inv (DOM f\<^bold>\<down>) \<^bold>\<cdot> \<^bold>\<lfloor>rep f\<^bold>\<rfloor> \<^bold>\<cdot> COD f\<^bold>\<down>)"
fold_insert: fixes B :: "'a set"   and x :: "'a"   and z :: "'a" assumes "finite B"   and "x \<notin> B"   and "insert x B \<subseteq> A"   and "z \<in> A" shows "Finite_Set.fold f z (insert x B) = f x (Finite_Set.fold f z B)"
monomials_monom: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'a" shows "monomials (monom m a) = (if a = (0::'a) then {} else {m})"
has_integral_iff: fixes f :: "'c \<Rightarrow> 'd"   and i :: "'d"   and S :: "'c set" shows "(f has_integral i) S = (f integrable_on S \<and> integral S f = i)"
HPhase1or2ReadThen_HInv4b: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc"   and r :: "Proc" assumes "HPhase1or2ReadThen s s' p d q"   and "HInv4b s r" shows "HInv4b s' r"
DEG_def: fixes to :: "'a nat_term_order" shows "DEG to = Abs_nat_term_order (deg_comp (nat_term_compare to))"
FOREACHc_refine: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and S' :: "'d set"   and \<sigma>0 :: "'e"   and \<sigma>0' :: "'f"   and R :: "('e \<times> 'f) set"   and \<Phi>'' :: "'c set \<Rightarrow> 'e \<Rightarrow> 'd set \<Rightarrow> 'f \<Rightarrow> bool"   and c :: "'e \<Rightarrow> bool"   and c' :: "'f \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e \<Rightarrow> 'e nres"   and f' :: "'d \<Rightarrow> 'f \<Rightarrow> 'f nres" assumes "inj_on \<alpha> S"   and "S' = \<alpha> ` S"   and "(\<sigma>0, \<sigma>0') \<in> R"   and "\<Phi>'' S \<sigma>0 (\<alpha> ` S) \<sigma>0'"   and "\<And>it \<sigma> it' \<sigma>'. \<lbrakk>it' = \<alpha> ` it; it \<subseteq> S; it' \<subseteq> S'; \<Phi>'' it \<sigma> it' \<sigma>'; (\<sigma>, \<sigma>') \<in> R\<rbrakk> \<Longrightarrow> c \<sigma> = c' \<sigma>'"   and "\<And>x it \<sigma> x' it' \<sigma>'. \<lbrakk>x' = \<alpha> x; x \<in> it; x' \<in> it'; it' = \<alpha> ` it; it \<subseteq> S; it' \<subseteq> S'; \<Phi>'' it \<sigma> it' \<sigma>'; c \<sigma>; c' \<sigma>'; (\<sigma>, \<sigma>') \<in> R\<rbrakk> \<Longrightarrow> f x \<sigma> \<le> \<Down> {(\<sigma>, \<sigma>'). (\<sigma>, \<sigma>') \<in> R \<and> \<Phi>'' (it - {x}) \<sigma> (it' - {x'}) \<sigma>'} (f' x' \<sigma>')" shows "FOREACH\<^sub>C S c f \<sigma>0 \<le> \<Down> R (FOREACH\<^sub>C S' c' f' \<sigma>0')"
deep_model_correct_params_def: fixes rs :: "nat list" shows "deep_model_correct_params rs \<equiv> 3 \<le> length rs \<and> (\<forall>r. r \<in> set rs \<longrightarrow> 0 < r)"
l_span_zero: fixes A :: "'b set" assumes "ideal R A" shows "linear_span R M A {\<zero>} = {\<zero>}"
simps_3: fixes uv :: "'b"   and t :: "'c tree" shows "U (Insert uv) [t] = 4 * \<phi> t + 3"
notSmallerTransaux: fixes x :: "('b, 'c) Combinators"   and r :: "('b, 'c) Combinators"   and y :: "('b, 'c) Combinators"   and l :: "'b set list" assumes "x \<noteq> DenyAll"   and "r \<noteq> DenyAll"   and "singleCombinators [x]"   and "singleCombinators [y]"   and "singleCombinators [r]"   and "\<not> smaller y x l"   and "smaller x y l"   and "smaller x r l"   and "smaller y r l"   and "in_list x l"   and "in_list y l"   and "in_list r l" shows "\<not> smaller r x l"
nested_rely: fixes j :: "'a::refinement_lattice"   and c :: "'a::refinement_lattice"   and i :: "'a::refinement_lattice" assumes "chaos \<sqsubseteq> (j::'a::refinement_lattice)" shows "(c::'a::refinement_lattice) // (j::'a::refinement_lattice) // (i::'a::refinement_lattice) = c // (i \<parallel> j)"
consistent_nonneg: shows "0 \<le> euler_C TYPE('a)"
integrate_insert_commute_delete_axioms_def: fixes M :: "('a, 'b) message set"   and m :: "'a delete_message" shows "integrate_insert_commute_delete_axioms M m \<equiv> consistent (M \<union> {Delete m})"
image_Nset_Suc: fixes f :: "nat \<Rightarrow> 'c"   and n :: "nat" shows "f ` {j. j \<le> Suc n} = insert (f (Suc n)) (f ` {j. j \<le> n})"
a: shows "steps (s, l, r) (shift (findnth n) (start_of ly as - Suc 0) @ adjust (shift (shift tdec_b (2 * n)) (start_of ly as - Suc 0)) (start_of ly e), start_of ly as - Suc 0) stpa = (start_of ly as + 2 * n, la, ra) \<and> inv_locate_a (as, lm) (n, la, ra) ires"
partition_rev_filter_conv: fixes P :: "'a \<Rightarrow> bool"   and yes :: "'a list"   and no :: "'a list"   and xs :: "'a list" shows "partition_rev P (yes, no) xs = (rev (filter P xs) @ yes, rev (filter (Not \<circ> P) xs) @ no)"
simps_3: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and xs :: "'a list" shows "binrelchain P (x # y # xs) = (P x y \<and> binrelchain P (y # xs))"
moebius_pt_moebius_similarity: fixes a :: "complex"   and b :: "complex"   and z :: "complex_homo" assumes "a \<noteq> 0" shows "moebius_pt (moebius_similarity a b) z = of_complex a *\<^sub>h z +\<^sub>h of_complex b"
cring_simprules_12: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "cring R"   and "x \<in> carrier R" shows "\<one>\<^bsub>R\<^esub> \<otimes>\<^bsub>R\<^esub> x = x"
add_0_right: fixes a :: "'b" shows "a \<sqinter> top = a"
the_all_deriv: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'b \<Rightarrow> 'c" assumes "\<forall>x. (f has_derivative F x) (at x)" shows "(THE G. \<forall>x. (f has_derivative G x) (at x)) = F"
isos_compose: fixes f :: "'a"   and f' :: "'a" assumes "S.iso f"   and "S.iso f'"   and "S.seq f' f" shows "S.iso (f' \<cdot>\<^sub>S f)"
list_succ_orbit_list: fixes s :: "'b"   and f :: "'b \<Rightarrow> 'b" assumes "s \<in> orbit f s"   and "\<And>x. x \<notin> orbit f s \<Longrightarrow> f x = x" shows "list_succ (orbit_list f s) = f"
FL_bisimilar_transp: fixes F :: "'effect fs_set" shows "transp (FL_bisimilar F)"
SeqHR_def: fixes x :: "hf"   and x' :: "hf"   and s :: "hf"   and k :: "hf" shows "SeqHR x x' s k \<equiv> BuildSeq2 (\<lambda>y y'. Ord y \<and> WR y y') (\<lambda>u u' v v' w w'. u = \<langle>v, w\<rangle> \<and> u' = q_HPair v' w') s k x x'"
lim: shows "AE x in M. ((\<lambda>i. s i x) \<longlongrightarrow> f x) at_top"
inject_2: fixes x2 :: "'b"   and y2 :: "'b" shows "(Inr x2 = Inr y2) = (x2 = y2)"
foundation18': fixes \<tau> :: "'a state \<times> 'a state"   and X :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "(\<tau> \<Turnstile> \<upsilon> X) = (X \<tau> \<noteq> UML_Types.bot_class.bot)"
elims: fixes x :: "role_t"   and y :: "Runs.var set" assumes "domain x = y" obtains "x = Init" and "y = {xnx, xgnx, xgny, xsk, xEnd}"   | "x = Resp" and "y = {xny, xgnx, xgny, xsk, xEnd}"
not_def: fixes P :: "bool" shows "\<not> P \<equiv> P \<longrightarrow> False"
A42_NSources_L1: shows "\<forall>C\<in>AbstrLevel level1. C \<noteq> sA5 \<longrightarrow> sA42 \<notin> Sources level1 C"
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>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: fixes ptr :: "(_) element_ptr" shows "cast ptr = node_ptr.Ext (Inl ptr)"
eq_take_length_prefix: fixes xs :: "[:'a:]"   and ys :: "[:'a:]" shows "prefix\<cdot>xs\<cdot>ys = eq\<cdot>xs\<cdot>(stake\<cdot>(slength\<cdot>xs)\<cdot>ys)"
rep_eq: fixes x :: "('c, 'd) lifted_pair" shows "Rep_lifted_pair (- x) = (\<lambda>f. triple.pairs_uminus (Rep_phi f) (Rep_lifted_pair x f))"
False: shows "\<not> (f = 0 \<or> g = 0)"
lappend_inf: fixes xs :: "'a llist"   and ys :: "'a llist" assumes "\<not> lfinite xs" shows "lappend xs ys = xs"
entries_rbt_lookup: fixes t1 :: "('a, 'c) rbt"   and t2 :: "('a, 'c) rbt" assumes "rbt_sorted t1"   and "rbt_sorted t2" shows "(entries t1 = entries t2) = (rbt_lookup t1 = rbt_lookup t2)"
case_distrib: fixes h :: "'d \<Rightarrow> 'c"   and f1 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> k \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool \<Rightarrow> 'd"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'd"   and f3 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat \<Rightarrow> 'd"   and f4 :: "nat \<Rightarrow> nat \<Rightarrow> 'd"   and f5 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'd"   and msg :: "msg" shows "h (case msg of Rreq x xa xb xc xd xe xf xg \<Rightarrow> f1 x xa xb xc xd xe xf xg | Rrep x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd | Rerr x xa \<Rightarrow> f3 x xa | Newpkt x xa \<Rightarrow> f4 x xa | Pkt x xa xb \<Rightarrow> f5 x xa xb) = (case msg of Rreq x1 x2 x3 x4 x5 x6 x7 x8 \<Rightarrow> h (f1 x1 x2 x3 x4 x5 x6 x7 x8) | Rrep x1 x2 x3 x4 x5 \<Rightarrow> h (f2 x1 x2 x3 x4 x5) | Rerr x1 x2 \<Rightarrow> h (f3 x1 x2) | Newpkt x1 x2 \<Rightarrow> h (f4 x1 x2) | Pkt x1 x2 x3 \<Rightarrow> h (f5 x1 x2 x3))"
insertion_monom: fixes f :: "nat \<Rightarrow> 'a"   and m :: "nat \<Rightarrow>\<^sub>0 nat"   and c :: "'a" shows "insertion f (MPoly_Type.monom m c) = c * (\<Prod>x. f x ^ lookup m x)"
arr_dom_iff_arr: fixes f :: "'c local.term set" shows "\<F>C.arr (\<F>C.dom f) = \<F>C.arr f"
wordinterval_CIDR_split_prefixmatch_all_valid_Ball: fixes r :: "'b wordinterval" shows "\<forall>e\<in>set (wordinterval_CIDR_split_prefixmatch r). valid_prefix e \<and> pfxm_length e \<le> LENGTH('b)"
shrK_in_initState: fixes A :: "agent" shows "Key (shrK A) \<in> initState A"
is_tangle_diagram_matrix_match: fixes w1 :: "brick list"   and ws1 :: "wall"   and w2 :: "brick list"   and ws2 :: "wall" assumes "is_tangle_diagram (w1 * ws1)"   and "is_tangle_diagram (w2 * ws2)" shows "rat_poly.matrix_match (blockmat w1) (kauff_mat ws1) (blockmat w2) (kauff_mat ws2)"
real_alg_show_info_def: fixes x :: "real_alg" shows "real_alg_show_info x = convert_info (info_real_alg x)"
rep_eq: fixes x :: "('c, 'd) sq_matrix" shows "to_vec x = (\<chi>i. vec_lambda (to_fun x i))"
subseqs_distinctD: fixes ys :: "'a list"   and xs :: "'a list" assumes "ys \<in> set (subseqs xs)"   and "distinct xs" shows "distinct ys"
caller_def: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> ('c, 'd) chan" shows "caller ch \<equiv> \<lambda>s p. (cbit (ch s p), arg (ch s p))"
enat_leI: fixes m :: "enat"   and n :: "enat"   and X :: "(enat \<times> enat) set" assumes "(m, n) \<in> X"   and "\<And>m n. (m, n) \<in> X \<Longrightarrow> m = 0 \<or> (\<exists>m' n' k. m = eSuc m' \<and> n = n' + enat k \<and> k \<noteq> 0 \<and> ((m', n') \<in> X \<or> m' \<le> n'))" shows "m \<le> n"
TE_Tag: fixes f :: "'fsym"   and al :: "univ list" assumes "wtFsym f"   and "list_all2 eintT (arOf f) al" shows "eintF (Tag (resOf f)) [eintF (Oldf f) al] = eintF (Oldf f) al"
indep_in_carrier: shows "\<E>.indep_in carrier = indep_in carrier"
that_2: assumes "u \<notin> R"   and "(u, v) \<in> (rel_restrict E R)\<^sup>*" shows "thesis"
dim: fixes n :: "nat" assumes "Nd \<le> n" shows "domain (P n) = d"
invrules_3: shows "bij_exists P\<^sub>1 (\<Union> (P\<^sub>2 \<union> wrap B\<^sub>2))"
integrable_on_subinterval: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set"   and a :: "'b"   and b :: "'b" assumes "f integrable_on S"   and "{a..b} \<subseteq> S" shows "f integrable_on {a..b}"
lookup_except_eq_idI: fixes t :: "'c"   and S :: "'c set"   and p :: "'c \<Rightarrow>\<^sub>0 'd" assumes "t \<notin> S" shows "lookup (except p S) t = lookup p t"
hyp: fixes env :: "('a, 'b, 'c) env"   and e :: "('a, 'b, 'c) env" assumes "lookup env xs = Some e" shows "lookup (update xs (Some env') env) xs = Some env'"
sigma_algebra_sigma_sets: fixes a :: "'a set set"   and \<Omega> :: "'a set" assumes "a \<subseteq> Pow \<Omega>" shows "sigma_algebra \<Omega> (sigma_sets \<Omega> a)"
idem: fixes a :: "'b::semilattice_inf" shows "(a::'b::semilattice_inf) \<sqinter> a = a"
mk_minsky_zero: fixes v :: "nat" shows "mk_minsky (\<lambda>vs vs'. vs' = (\<lambda>x. if x = v then 0 else vs x))"
1: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) set" shows "hom_induced p (subtopology X S) T X T id x \<in> kernel (relative_homology_group p X T) (relative_homology_group p X S) (hom_induced p X T X S id)"
simps_8: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case Lt of Eq \<Rightarrow> f1 | Lt \<Rightarrow> f2 | Gt \<Rightarrow> f3) = f2"
prv_eqv_imp_transi: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and \<phi>1 :: "'c"   and \<phi>2 :: "'c"   and \<chi> :: "'c" assumes "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<chi> \<in> fmla"   and "prv (Syntax_with_Connectives.eqv cnj imp \<phi>1 \<phi>2)"   and "prv (imp \<phi>2 \<chi>)" shows "prv (imp \<phi>1 \<chi>)"
runit_char_2: fixes f :: "'a" assumes "ide f" shows "\<guillemotleft>PB.runit f : f \<star> src f \<Rightarrow> f\<guillemotright>"
simps: fixes p1 :: "(action \<times> nat) list"   and p2 :: "(action \<times> nat) list" shows "p1 |+| p2 = adjust0 p1 @ shift p2 (length p1 div 2)"
gtrI_s: fixes s :: "'c"   and c :: "'c multiset"   and e :: "'d"   and s' :: "'e"   and c' :: "'e multiset"   and gtrs :: "(('c \<times> 'c multiset) \<times> 'd \<times> 'e \<times> 'e multiset) set" assumes "((s, c), e, s', c') \<in> gtrs" shows "(add_mset s c, e, add_mset s' c') \<in> gtr gtrs"
rref_upt_condition3: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat" assumes "reduced_row_echelon_form_upt_k A k" shows "\<forall>i. i < i + (1::'c) \<and> \<not> is_zero_row_upt_k i k A \<and> \<not> is_zero_row_upt_k (i + (1::'c)) k A \<longrightarrow> (LEAST n. A $ i $ n \<noteq> (0::'a)) < (LEAST n. A $ (i + (1::'c)) $ n \<noteq> (0::'a))"
lm029: fixes x :: "'c"   and X :: "'c set" shows "(x \<in> X) = ({x} \<in> finestpart X)"
subtype_refl: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and t :: "ty" shows "subtype P t t"
relpowp_Suc_I2: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and n :: "nat"   and z :: "'c" assumes "P x y"   and "(P ^^ n) y z" shows "(P ^^ Suc n) x z"
assms_1: shows "\<langle>x\<rangle> \<Turnstile>\<^sub>c\<^sub>s set (vars_from_index_geq_vec start b)"
representation_zero: fixes basis :: "'d set" shows "crepresentation basis (0::'d) = (\<lambda>b. 0)"
size_eq_sum_mset: fixes M :: "'b multiset" shows "size M = (\<Sum>a\<in>#M. 1)"
norm_equiv: fixes x :: "'b itrm" shows "norm x \<simeq> x"
recurrent_iff_G_infinite: fixes K :: "'a \<Rightarrow> 'a pmf"   and x :: "'a" shows "MC_syntax.recurrent K x = (MC_syntax.G K x x = \<infinity>)"
rel2p_inv_2: fixes R :: "('a \<times> 'b) set" shows "p2rel (rel2p R) = R"
pseudofunctor_axioms: shows "pseudofunctor (\<cdot>\<^sub>B) (\<star>\<^sub>B) \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<cdot>) (\<star>) \<a> \<i> src trg UP cmp\<^sub>U\<^sub>P"
eq_elems_eq_val: fixes x :: "'d"   and y :: "'d"   and f :: "'d \<Rightarrow> 'e" assumes "x = y" shows "f x = f y"
enat_defs_4: fixes m :: "enat"   and n :: "enat" shows "m + n = (case m of enat m \<Rightarrow> case n of enat n \<Rightarrow> enat (m + n) | \<infinity> \<Rightarrow> \<infinity> | \<infinity> \<Rightarrow> \<infinity>)"
filter_eq_Cons_iff: fixes P :: "'d \<Rightarrow> bool"   and ys :: "'d list"   and x :: "'d"   and xs :: "'d list" shows "(filter P ys = x # xs) = (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
hfrefD: fixes f :: "'a \<Rightarrow> 'b Heap"   and g :: "'c \<Rightarrow> 'd nres"   and P :: "'c \<Rightarrow> bool"   and RS :: "('c \<Rightarrow> 'a \<Rightarrow> assn) \<times> ('c \<Rightarrow> 'a \<Rightarrow> assn)"   and T :: "'d \<Rightarrow> 'b \<Rightarrow> assn"   and a :: "'c"   and c :: "'a" assumes "(f, g) \<in> [P]\<^sub>a RS \<rightarrow> T"   and "P a" shows "hn_refine (fst RS a c) (f c) (snd RS a c) T (g a)"
prems_2: fixes x :: "'a"   and y :: "'a" assumes "x \<in> set (map fst (pair # pairs))"   and "y \<in> set (map fst (pair # pairs))"   and "P x"   and "P y" shows "x = y"
schroeder_1_var: fixes x :: "'a" shows "is_conjugation ((;) x) ((;) (x\<^sup>T))"
invertible_fst_Gauss_Jordan_column_k_PA: fixes P :: "(('a, 'b) vec, 'b) vec"   and i :: "nat"   and A :: "(('a, 'c) vec, 'b) vec"   and k :: "nat" assumes "invertible P" shows "invertible (fst (Gauss_Jordan_column_k_PA (P, i, A) k))"
rScope: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and x :: "name"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P'"   and "\<And>C. Prop C \<Psi> P \<alpha> P'"   and "x \<sharp> \<Psi>"   and "x \<sharp> \<alpha>"   and "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "x \<sharp> C"   and "bn \<alpha> \<sharp>* C"   and "distinct (bn \<alpha>)" shows "Prop C \<Psi> (\<lparr>\<nu>x\<rparr>P) \<alpha> (\<lparr>\<nu>x\<rparr>P')"