hcju/isabelleps huggingface dataset:
assms_1: shows "wbt \<langle>l, (a, n), r\<rangle>"
del_locs_def: fixes s :: "state" shows "del_locs s \<equiv> s\<lparr>locals := Map.empty\<rparr>"
Empty_correct_2: shows "FingerTreeStruc.ft_invar Empty"
Refl: fixes P :: "pi" shows "P \<equiv>\<^sub>e P"
con_rews_1: fixes slist :: "[:'a:]" shows "\<bottom> :# slist = \<bottom>"
abs_eq: fixes x :: "'a \<Rightarrow> 'b" assumes "eq_onp (\<lambda>f. finite {x. f x \<noteq> (0::'b)}) x x" shows "keys (Abs_poly_mapping x) = {k. x k \<noteq> (0::'b)}"
mult_fract_cancel: fixes c :: "'b"   and a :: "'b"   and b :: "'b" assumes "c \<noteq> (0::'b)" shows "Fract (c * a) (c * b) = Fract a b"
V: fixes y :: "'c" assumes "y \<in> U" shows "y \<in> V y"
RID\<^sub>GI_2: fixes s :: "'e \<Rightarrow> (('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('e, 'd, 'f) expr) option"   and r' :: "'e"   and ls :: "('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('e, 'd, 'f) expr"   and r :: "'e" assumes "s r' = Some ls"   and "r \<in> RID\<^sub>L ls" shows "r \<in> RID\<^sub>G s"
choose_v_def: fixes G :: "nat set \<times> nat set set"   and k :: "nat" shows "choose_v G k \<equiv> SOME u. \<exists>p. p \<in> short_cycles G k \<and> u \<in> set p"
intra_SDG_path_matched: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node" assumes "n i-ns\<rightarrow>\<^sub>d* n'" shows "matched n ns n'"
finite_succeed_t'': fixes i :: "nat" assumes "q\<^sub>0 \<in> F" shows "finite (succeed i) = finite (succeed_t i)"
simps_1: fixes ys :: "'a list" shows "splice [] ys = ys"
is_pappus2_def: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and Q :: "Points"   and R :: "Points" shows "is_pappus2 A B C A' B' C' P Q R \<equiv> distinct6 A B C A' B' C' \<or> A \<noteq> B' \<and> A' \<noteq> B \<and> line A B' \<noteq> line A' B \<and> B \<noteq> C' \<and> B' \<noteq> C \<and> line B C' \<noteq> line B' C \<and> A \<noteq> C' \<and> A' \<noteq> C \<and> line A C' \<noteq> line A' C \<longrightarrow> col A B C \<longrightarrow> col A' B' C' \<longrightarrow> is_a_intersec P A B' A' B \<longrightarrow> is_a_intersec Q B C' B' C \<longrightarrow> is_a_intersec R A C' A' C \<longrightarrow> col P Q R"
uminus_uint64_def: shows "uminus \<equiv> map_fun Rep_uint64 Abs_uint64 uminus"
SXcpt_subcls_Throwable_lemma: fixes G :: "prog"   and xn :: "xname"   and xc :: "class" assumes "class G (SXcpt xn) = Some xc"   and "super xc = (if xn = Throwable then Object else SXcpt Throwable)" shows "G\<turnstile>SXcpt xn\<preceq>\<^sub>C SXcpt Throwable"
Gromov_boundary_extended_product_PInf: fixes e :: "'b"   and x :: "'b Gromov_completion"   and y :: "'b Gromov_completion" shows "(extended_Gromov_product_at e x y = \<infinity>) = (x \<in> Gromov_boundary \<and> y = x)"
balance_right_inv1: fixes a :: "('a, 'b) rbt"   and b :: "('a, 'b) rbt"   and k :: "'a"   and x :: "'b" assumes "inv1 a"   and "inv1l b"   and "color_of a = B" shows "inv1 (balance_right a k x b)"
assms: shows "h # slist \<in> state_set (state_list s as)"
asympI: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>a b. R a b \<Longrightarrow> \<not> R b a" shows "asymp R"
distinct_25: fixes x3 :: "nat"   and x5 :: "msg" shows "Nonce x3 \<noteq> Hash x5"
\<delta>\<^sub>p_def_2: shows "(\<delta>\<^sub>p', p\<^sub>1') = find_closest_bf_code p\<^sub>0 (take 7 (p\<^sub>2 # ps))"
simps_6: fixes e1 :: "expr"   and e2 :: "expr" shows "expr_rf_to_cexpr (LET e1 IN e2) = cexpr_subst 0 (expr_rf_to_cexpr e1) (expr_rf_to_cexpr e2)"
divides_mult_lI: fixes a :: "'a list"   and b :: "'a list"   and c :: "'a list" assumes "a divides\<^bsub>poly_ring R\<^esub> b"   and "a \<in> carrier (poly_ring R)"   and "c \<in> carrier (poly_ring R)" shows "c \<otimes>\<^bsub>poly_ring R\<^esub> a divides\<^bsub>poly_ring R\<^esub> c \<otimes>\<^bsub>poly_ring R\<^esub> b"
gkern1: fixes F :: "('e, 'f) Group_scheme"   and E :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g" assumes "Group F"   and "Ugp E"   and "f \<in> gHom F E" shows "gker\<^bsub>F,E\<^esub> f = carrier F"
simps_11: fixes v :: "event_data list"   and y :: "nat"   and \<omega> :: "agg_type \<times> event_data"   and b :: "nat"   and f :: "trm"   and \<phi> :: "formula"   and e :: "String.literal \<times> event_data list" shows "matches v (Agg y \<omega> b f \<phi>) e = (\<exists>zs. length zs = b \<and> matches (zs @ v) \<phi> e)"
ereal_LimI_finite: fixes x :: "ereal"   and u :: "nat \<Rightarrow> ereal" assumes "\<bar>x\<bar> \<noteq> \<infinity>"   and "\<And>r. 0 < r \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. u n < x + r \<and> x < u n + r" shows "u \<longlonglongrightarrow> x"
apfst_def: fixes f :: "'b \<Rightarrow> 'd" shows "apfst f = map_prod f id"
UU: shows "S \<inter> f -` f ` (S \<inter> U) = S \<inter> U"
h5: shows "MPoly_Type.degree (f i) x \<le> i"
2: fixes u :: "'a list"   and a :: "'a"   and v :: "'a list"   and w' :: "'a stream" assumes "w = u @- a ## v @- a ## w'" shows "thesis"
split: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'c"   and f2 :: "'e \<Rightarrow> 'c"   and RHS :: "('d, 'e) RHS" shows "P (case RHS of Branch x xa \<Rightarrow> f1 x xa | Leaf x \<Rightarrow> f2 x) = ((\<forall>x11 x12. RHS = Branch x11 x12 \<longrightarrow> P (f1 x11 x12)) \<and> (\<forall>x2. RHS = Leaf x2 \<longrightarrow> P (f2 x2)))"
sl6: shows "a\<^sup>2 * e\<^sup>2 + f\<^sup>2 + - 2 * e * b * f + b\<^sup>2 * e\<^sup>2 + - c * e\<^sup>2 \<in> radical_sqrt"
cong: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "\<forall>\<^sub>F x in F. f x = g x" shows "O[F](f) = O[F](g)"
IH: assumes "distinct list" shows "succ_rel (insert_spec list (oid, Some ref)) = insert_alt (succ_rel list) (oid, ref)"
neg_numeral_neq_numeral: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and m :: "num"   and n :: "num" assumes "class.ring_char_0 one times plus zero minus uminus" shows "uminus (numeral.numeral one plus m) \<noteq> numeral.numeral one plus n"
perm_3: fixes pi :: "(name \<times> name) list"   and x1 :: "name"   and x2 :: "lam" shows "pi \<bullet> Lam [x1].x2 = Lam [(pi \<bullet> x1)].(pi \<bullet> x2)"
height_list_split: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and ls :: "('c btree \<times> 'c) list"   and a :: "'c btree"   and b :: "'c"   and rs :: "('c btree \<times> 'c) list"   and t :: "'c btree" assumes "split split" shows "height_up\<^sub>i (Up\<^sub>i (Node ls a) b (Node rs t)) = height (Node (ls @ (a, b) # rs) t)"
trapG_def: shows "trapG = trapG1 \<inter> {p. p1 (fst p) (snd p) \<le> 0}"
SUP_insert: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and f :: "'b \<Rightarrow> 'a"   and a :: "'b"   and A :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (f ` insert a A) = sup (f a) (Sup (f ` A))"
h1: shows "n = a\<^sup>2 + b\<^sup>2"
generate_in_carrier: fixes H :: "'a set"   and h :: "'a" assumes "H \<subseteq> carrier R"   and "h \<in> generate (add_monoid R) H" shows "h \<in> carrier R"
a_subid': fixes x :: "'a" shows "ar x \<le> (1::'a)"
Trans: shows "\<Psi> \<rhd> P \<parallel> Q \<longmapsto> \<alpha> \<prec> R"
rep_eq: fixes "next" :: "'b \<Rightarrow> 'b"   and arb :: "'b"   and x :: "'b fset" assumes "fresh next" shows "fresh.fNext next arb x = fresh.Next next arb (fset x)"
periodic: fixes x :: "real" shows "f (x + 2 * pi) = f x"
csum_def: fixes r1 :: "('d \<times> 'd) set"   and r2 :: "('e \<times> 'e) set" shows "r1 +c r2 \<equiv> |Field r1 <+> Field r2|"
freshAbs_simp: fixes X :: "('b, 'c, 'varSort, 'var, 'd) qTerm set"   and ys :: "'varSort"   and y :: "'var"   and xs :: "'varSort"   and x :: "'var" assumes "good X" shows "freshAbs ys y (Abs xs x X) = (ys = xs \<and> y = x \<or> fresh ys y X)"
lfp_transfer_bounded: fixes P :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b"   and \<alpha> :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'c" assumes "P bot"   and "\<And>x. P x \<Longrightarrow> P (f x)"   and "\<And>M. (\<And>i. P (M i)) \<Longrightarrow> P (Sup (range M))"   and "\<And>M. \<lbrakk>incseq M; \<And>i. P (M i)\<rbrakk> \<Longrightarrow> \<alpha> (Sup (range M)) = (SUP i. \<alpha> (M i))"   and "sup_continuous f"   and "sup_continuous g"   and "\<And>x. \<lbrakk>P x; x \<le> lfp f\<rbrakk> \<Longrightarrow> \<alpha> (f x) = g (\<alpha> x)"   and "\<And>x. \<alpha> bot \<le> g x" shows "\<alpha> (lfp f) = lfp g"
N_1: fixes n :: "nat" assumes "N \<le> n" shows "3 * alpha < Sigma_Algebra.measure (distr M borel (\<lambda>x. birkhoff_sum f n x / B n)) {x. a < \<bar>x\<bar>}"
fineI: fixes s :: "('c \<times> 'd set) set"   and d :: "'c \<Rightarrow> 'd set" assumes "\<And>x k. (x, k) \<in> s \<Longrightarrow> k \<subseteq> d x" shows "d fine s"
triangular_to_def: fixes k :: "nat"   and A :: "'a mat" shows "triangular_to k A \<equiv> \<forall>j<k. triangular_column j A"
delete_edges_def: fixes G :: "'b graph"   and es :: "('b \<times> 'b) set" shows "FiniteGraph.delete_edges G es \<equiv> \<lparr>nodes = nodes G, edges = {(e1, e2). (e1, e2) \<in> edges G \<and> (e1, e2) \<notin> es}\<rparr>"
eventually_all_finite2: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and F :: "'c filter" assumes "\<forall>i. eventually (P i) F" shows "\<forall>\<^sub>F x in F. \<forall>i. P i x"
coset_join1: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and x :: "'c" assumes "Group.group G"   and "H #>\<^bsub>G\<^esub> x = H"   and "x \<in> carrier G"   and "subgroup H G" shows "x \<in> H"
SourcesS6_L2: shows "Sources level2 sS6 = {sS2, sS4, sS5}"
W_def: shows "W \<equiv> {x \<in> U. setdist {x} S = setdist {x} T}"
disj_def: fixes x :: "Value"   and y :: "Value"   and s :: "Store" shows "StoreProperties.disj x y s = (\<forall>l. \<not> s\<turnstile> l reachable_from x \<or> \<not> s\<turnstile> l reachable_from y)"
long_division_closed_1: fixes K :: "'a list set"   and p :: "'a list list"   and q :: "'a list list" assumes "subfield K (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "q \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)" shows "UP.pdiv p q \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"
infnorm_set_image: fixes x :: "'a" shows "{\<bar>x \<bullet> i\<bar> |i. i \<in> Basis} = (\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis"
crdt_ops_distinct: fixes xs :: "('c \<times> 'd) list"   and deps :: "'d \<Rightarrow> 'c set" assumes "crdt_ops xs deps" shows "distinct xs"
collapse_5: fixes rexp :: "'b rexp" assumes "is_Times rexp" shows "Times (un_Times1 rexp) (un_Times2 rexp) = rexp"
monoid_axioms: shows "monoid (\<cdot>) (1::'e)"
RInit\<^sub>1: fixes P :: "nat exp prog"   and e :: "nat exp"   and s :: "heap \<times> val list \<times> sheap"   and v :: "val"   and h' :: "heap"   and l' :: "val list"   and sh' :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and i :: "init_state"   and sh'' :: "sheap"   and C' :: "char list"   and Cs :: "char list list"   and e' :: "nat exp"   and e\<^sub>1 :: "nat exp"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<rangle> \<Rightarrow> \<langle>Val v,(h', l', sh')\<rangle>"   and "sh' C = \<lfloor>(sfs, i)\<rfloor>"   and "sh'' = sh'(C \<mapsto> (sfs, Done))"   and "C' = last (C # Cs)"   and "P \<turnstile>\<^sub>1 \<langle>INIT C' (Cs,True) \<leftarrow> e',(h', l', sh'')\<rangle> \<Rightarrow> \<langle>e\<^sub>1,s\<^sub>1\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>RI (C,e) ; Cs \<leftarrow> e',s\<rangle> \<Rightarrow> \<langle>e\<^sub>1,s\<^sub>1\<rangle>"
comb_sat: shows "1 < length qs \<longrightarrow> p \<noteq> 0 \<and> 0 < length qs \<longrightarrow> satisfies_properties_R p qs (get_subsets_R (snd (combine_systems_R p (take (length qs div 2) qs, calculate_data_R p (take (length qs div 2) qs)) (drop (length qs div 2) qs, calculate_data_R p (drop (length qs div 2) qs))))) (get_signs_R (snd (combine_systems_R p (take (length qs div 2) qs, calculate_data_R p (take (length qs div 2) qs)) (drop (length qs div 2) qs, calculate_data_R p (drop (length qs div 2) qs))))) (get_matrix_R (snd (combine_systems_R p (take (length qs div 2) qs, calculate_data_R p (take (length qs div 2) qs)) (drop (length qs div 2) qs, calculate_data_R p (drop (length qs div 2) qs)))))"
prv_all_gen: fixes x :: "'var"   and \<phi> :: "'fmla" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "prv \<phi>" shows "prv (all x \<phi>)"
radical_def: fixes F :: "'c set" shows "\<surd>F = {f. \<exists>m. f ^ m \<in> F}"
singletonI: fixes a :: "'c" shows "a \<in> \<eta> a"
splitset_eq_splitset_fun: fixes xs :: "'c list" shows "splitset xs = splitset_fun xs"
det_fold_map_def: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and c :: "'e \<Rightarrow> bool"   and f :: "'d \<times> 'f \<Rightarrow> 'e \<Rightarrow> 'e"   and \<sigma> :: "'e"   and result :: "('d \<Rightarrow> 'f option) \<Rightarrow> 'e" shows "det_fold_map R c f \<sigma> result \<equiv> \<forall>l. distinct (map fst l) \<and> sorted_wrt (key_rel R) l \<longrightarrow> foldli l c f \<sigma> = result (map_of l)"
subst_vsubst_compose1: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and y :: "'var"   and y1 :: "'var"   and ys :: "'varSort" assumes "good X"   and "good Y"   and "y \<noteq> y1" shows "X #[y1 // y]_ys #[Y / y]_ys = X #[y1 // y]_ys"
iso_arcs_eq_iff: fixes G :: "('e, 'f) pre_digraph"   and hom :: "('e, 'f, 'g, 'h) digraph_isomorphism"   and e1 :: "'f"   and e2 :: "'f" assumes "pre_digraph.digraph_isomorphism G hom"   and "e1 \<in> arcs G"   and "e2 \<in> arcs G" shows "(iso_arcs hom e1 = iso_arcs hom e2) = (e1 = e2)"
abc_Hoare_plus_unhalt1: fixes P :: "nat list \<Rightarrow> bool"   and A :: "abc_inst list"   and B :: "abc_inst list" assumes "{P} A \<up>" shows "{P} A [+] B \<up>"
insert_or: fixes t1 :: "'a"   and P :: "'a \<Rightarrow> bool" shows "{s. s = t1 \<or> P s} = insert t1 {s. P s}"
sum: fixes x :: "'a"   and g :: "'h \<Rightarrow> 'b"   and S :: "'h set" shows "h x (sum g S) = (\<Sum>a\<in>S. h x (g a))"
assms_3: fixes b :: "real" assumes "a \<le> b" shows "set_integrable lborel {a..<b} f"
is_processT5_S7': fixes t :: "'c event list"   and X :: "'c event set"   and P :: "'c process"   and A :: "'c event set" assumes "(t, X) \<in> F P"   and "(t, X \<union> A) \<notin> F P" shows "\<exists>x. x \<in> A \<and> x \<notin> X \<and> t @ [x] \<in> T P"
cases: fixes a :: "'b stream" assumes "ssorted a" obtains s :: "'b stream" where "a = s" and "shd s \<le> shd (stl s)" and "ssorted (stl s)"
same_suffix_suffix: fixes ys :: "'b list"   and xs :: "'b list"   and zs :: "'b list" shows "suffix (ys @ xs) (zs @ xs) = suffix ys zs"
c5: assumes "\<pi>\<^sub>3 r = unk" shows "P (rt(ip \<mapsto> (\<pi>\<^sub>2 (the (rt ip)), \<pi>\<^sub>3 r, \<pi>\<^sub>4 r, \<pi>\<^sub>5 r, \<pi>\<^sub>6 r, \<pi>\<^sub>7 (addpre r (\<pi>\<^sub>7 (the (rt ip)))))))"
wd_repeat: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and n :: "nat" assumes "well_def a" shows "well_def (repeat n a)"
ek_def: shows "ek = (u - ec - eps) / sum_list' p [snd (trunc_bound_eucl p (a * fst X + b * fst Y + c)), snd (trunc_bound_pdevs p (add_pdevs (scaleR_pdevs a (snd X)) (scaleR_pdevs b (snd Y)))), d]"
subst_drop: fixes x :: "name"   and t :: "'a"   and env :: "(name, 'a) fmap" assumes "x |\<notin>| frees t" shows "subst t (fmdrop x env) = subst t env"
STL2: fixes F :: "behavior \<Rightarrow> bool" shows "\<turnstile> \<box>F \<longrightarrow> F"
subgroup_imp_group: fixes G :: "('d, 'e) monoid_scheme"   and H :: "'d set" assumes "Group.group G"   and "subgroup H G" shows "Group.group (G\<lparr>carrier := H\<rparr>)"
compact_diff: fixes "open" :: "'a set \<Rightarrow> bool"   and S :: "'a set"   and T :: "'a set" assumes "class.topological_space open"   and "topological_space.compact open S"   and "open T" shows "topological_space.compact open (S - T)"
Collect_mem_eq: fixes A :: "'b set" shows "{x. x \<in> A} = A"
resumption_ord_simps_1: fixes out :: "'e"   and c :: "'f \<Rightarrow> ('d, 'e, 'f) Resumption.resumption"   and r :: "('d, 'e, 'f) Resumption.resumption" shows "resumption_ord (Resumption.resumption.Pause out c) r = (\<exists>c'. r = Resumption.resumption.Pause out c' \<and> ((=) ===> resumption_ord) c c')"
get_pending_impl_def: fixes s :: "('e, 'f) simple_state_scheme" shows "get_pending_impl s \<equiv> ASSERT (ss_stack s \<noteq> []) \<bind> (\<lambda>_. let (u, Vs) = hd (ss_stack s) in if Vs = {} then RETURN (u, None, s) else SPEC (\<lambda>v. v \<in> Vs) \<bind> (\<lambda>v. let Vs = Vs - {v}; s = s\<lparr>ss_stack := (u, Vs) # tl (ss_stack s)\<rparr> in RETURN (u, Some v, s)))"
alphaAbs_preserves_qGoodAbs: fixes A :: "('b, 'c, 'varSort, 'var, 'd) qAbs"   and A' :: "('b, 'c, 'varSort, 'var, 'd) qAbs" assumes "A $= A'" shows "qGoodAbs A = qGoodAbs A'"
count_increase1: fixes p :: "nat \<Rightarrow> nat"   and m :: "nat"   and k :: "nat" assumes "p partitions m - k"   and "sum p {..m - k} = k" shows "(\<Sum>i\<le>m. p (i - 1)) = k"
surjE: fixes f :: "'b \<Rightarrow> 'a"   and y :: "'a" assumes "surj f" obtains x :: "'b" where "y = f x"
power_closed: fixes A :: "'a set"   and x :: "'a"   and n :: "nat" assumes "ring_closed A"   and "x \<in> A" shows "x ^ n \<in> A"
igSwapBinp_def: fixes MOD :: "('m, 'n, 'o, 'p, 'q, 'r, 's, 'l, 't) model_scheme"   and zs :: "'o"   and z1 :: "'r"   and z2 :: "'r"   and binp :: "'k \<Rightarrow> 'l option" shows "igSwapBinp MOD zs z1 z2 binp \<equiv> lift (igSwapAbs MOD zs z1 z2) binp"
size_eq_card_implies_surj_on: fixes A :: "'c set"   and B :: "'d set"   and N :: "nat multiset"   and f :: "'c \<Rightarrow> 'd" assumes "finite A"   and "finite B"   and "size N = card B"   and "f \<in> Twelvefold_Way_Entry10.functions_of A B N" shows "f ` A = B"
realk: shows "real k * (2 * pi) / real n < 2 * pi"
Completeness: fixes I :: "'a set"   and s :: "'a set" assumes "I \<noteq> {}"   and "finite I"   and "Kripke {s. \<exists>i\<in>I. i \<rightarrow>\<^sub>i* s} I \<turnstile> EF s" shows "\<exists>A. \<turnstile>A \<and> attack A = (I, s)"
counit_simps_4: shows "src\<^sub>D (D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g)) = trg\<^sub>D (F f)"
cod_comp: fixes g :: "('a, 'a set) Maps.arr"   and f :: "('a, 'a set) Maps.arr" assumes "Maps.seq g f" shows "Maps.cod (g \<odot> f) = Maps.cod g"
inv_solve_left: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "(a = \<ominus> b \<oplus> c) = (c = b \<oplus> a)"
span_superset: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and S :: "'c set" assumes "module scale" shows "S \<subseteq> module.span scale S"
dehomogenize_monomial: fixes x :: "'b"   and t :: "'b \<Rightarrow>\<^sub>0 nat"   and c :: "'c" shows "dehomogenize x (monomial c t) = monomial c (except t {x})"
ordLess_transitive: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set"   and r'' :: "('f \<times> 'f) set" assumes "r <o r'"   and "r' <o r''" shows "r <o r''"
absorb2: fixes a :: "'c"   and b :: "'c" assumes "a \<le> b" shows "a \<squnion> b = b"
C14_dual: fixes x :: "'b"   and n :: "nat" shows "x\<^sup>\<star> = x\<^bsub>0\<^esub>\<^bsup>n\<^esup> \<cdot> (x\<^bsup>n + 1\<^esup>)\<^sup>\<star>"
card_geq_ex_subset: fixes n :: "nat"   and A :: "'c set" assumes "n \<le> card A" obtains B :: "'c set" where "card B = n" and "B \<subseteq> A"
iTL_defs_1: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "iAll I P \<equiv> \<forall>t\<in>I. P t"
False: shows "\<nexists>y. snd (rep_nat_term y) = i"
F_def: shows "F = (\<lambda>n x. min (f x) (of_nat n) * indicator (Y n) x)"
vars_snd_rcv_strand: fixes S :: "('d, 'c) strand_step list" shows "vars\<^sub>s\<^sub>t S = \<Union> (set (map fv\<^sub>s\<^sub>n\<^sub>d S)) \<union> \<Union> (set (map fv\<^sub>r\<^sub>c\<^sub>v S)) \<union> \<Union> (set (map (fv\<^sub>e\<^sub>q assign) S)) \<union> \<Union> (set (map (fv\<^sub>e\<^sub>q check) S)) \<union> \<Union> (set (map fv\<^sub>i\<^sub>n\<^sub>e\<^sub>q S)) \<union> bvars\<^sub>s\<^sub>t S"
cSUP_subset_mono: 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 A :: "'c set"   and g :: "'c \<Rightarrow> 'b"   and B :: "'c set"   and f :: "'c \<Rightarrow> 'b" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "A \<noteq> {}"   and "preorder.bdd_above less_eq (g ` B)"   and "A \<subseteq> B"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq (f x) (g x)" shows "less_eq (Sup (f ` A)) (Sup (g ` B))"
simps_2: fixes \<phi> :: "'b TESL_atomic"   and \<Phi> :: "'b TESL_atomic list" shows "\<lbrakk>\<lbrakk> \<phi> # \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = \<lbrakk> \<phi> \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L \<inter> \<lbrakk>\<lbrakk> \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L"
fun_upds_append_drop: fixes xs :: "'a list"   and ys :: "'b list"   and m :: "'a \<Rightarrow> 'b option"   and zs :: "'a list" assumes "length xs = length ys" shows "m(xs @ zs [\<mapsto>] ys) = m(xs [\<mapsto>] ys)"
bdd_above_def: fixes A :: "'a set" shows "bdd_above A = (\<exists>M. \<forall>x\<in>A. x \<le> M)"
intros_6: fixes P :: "'a trm"   and P' :: "'a trm" assumes "P >> P'" shows "Fst P >> Fst P'"
prj_simps_1: shows "C.arr trg_trg.prj\<^sub>1"
simps_5: fixes m :: "('d, 'e, 'f) resumption"   and f :: "'f \<Rightarrow> ('d, 'e, 'f) resumption" assumes "\<not> is_Done m \<or> \<not> is_Done (f (result m))" shows "resume (bind_resumption m f) = (if is_Done m then resume (f (result m)) else (\<lambda>i. bind_resumption (resume m i) f))"
pow_comm_comm': fixes u :: "'b list"   and k :: "nat"   and v :: "'b list"   and l :: "nat" assumes "u \<^sup>@ Suc k = v \<^sup>@ Suc l" shows "u \<cdot> v = v \<cdot> u"
ln_Gamma_real_converges: fixes x :: "real" assumes "0 < x" shows "convergent (ln_Gamma_series x)"
exec_call_1': fixes \<sigma> :: "'a control_state_scheme"   and M :: "'c \<Rightarrow> 'a control_state_scheme \<Rightarrow> ('b \<times> 'a control_state_scheme) option"   and A\<^sub>1 :: "'a control_state_scheme \<Rightarrow> 'c"   and M' :: "'a control_state_scheme \<Rightarrow> (bool \<times> 'a control_state_scheme) option" assumes "exec_stop \<sigma>" shows "(\<sigma> \<Turnstile> call_1\<^sub>C M A\<^sub>1 ;- M') = (\<sigma> \<Turnstile> M')"
preserves_reflects_arr: fixes f :: "'c" shows "S.arr (\<Phi> f) = Cop.arr f"
Un_absorb2: fixes B :: "'g set"   and A :: "'g set" assumes "B \<subseteq> A" shows "A \<union> B = A"
cases: fixes r :: "('a, 'b) graph" obtains nodes :: "'a set" and edges :: "('a \<times> 'b \<times> 'a) set" where "r = \<lparr>nodes = nodes, edges = edges\<rparr>"
part_equiv_sym: fixes R :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "part_equiv R"   and "(x, y) \<in> R" shows "(y, x) \<in> R"
main_lemma_1: fixes \<Gamma> :: "(name \<times> ty) list"   and s :: "trm"   and t :: "trm"   and T :: "ty" assumes "\<Gamma> \<turnstile> s is t : T"   and "valid \<Gamma>" shows "\<Gamma> \<turnstile> s \<Leftrightarrow> t : T"