hcju/isabelleps huggingface dataset:
assoc'_in_hom'_2: fixes \<mu> :: "'b"   and \<nu> :: "'b"   and \<tau> :: "'b" assumes "D.arr \<mu>"   and "D.arr \<nu>"   and "D.arr \<tau>"   and "src\<^sub>D \<mu> = trg\<^sub>D \<nu>"   and "src\<^sub>D \<nu> = trg\<^sub>D \<tau>" shows "\<guillemotleft>\<a>\<^sub>D\<^sup>-\<^sup>1[\<mu>, \<nu>, \<tau>] : D.dom \<mu> \<star>\<^sub>D D.dom \<nu> \<star>\<^sub>D D.dom \<tau> \<Rightarrow>\<^sub>D (D.cod \<mu> \<star>\<^sub>D D.cod \<nu>) \<star>\<^sub>D D.cod \<tau>\<guillemotright>"
assms_1: shows "theta \<subseteq> ZOretrT (theta \<union> ZObisT)"
scale_one: fixes x :: "'b" shows "(1::'b) * x = x"
is_extensional: fixes f :: "'a arrow_of_spans_data" assumes "\<not> arr f" shows "src f = null"
envs_def: fixes p :: "prog" shows "envs p \<equiv> {\<E>. match \<E> (ctxt p)}"
less_eq_assn_def: fixes a :: "assn"   and b :: "assn" shows "a \<le> b \<equiv> a = inf a b"
Count: shows "\<turnstile> Count  \<lbrace>\<acute>Proper \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>Ma \<and> Blacks \<acute>Ma \<subseteq> \<acute>bc \<and> \<acute>bc \<subseteq> Blacks \<acute>M \<and> length \<acute>Ma = length \<acute>M \<and> (\<acute>obc \<subset> Blacks \<acute>Ma \<or> \<acute>Safe)\<rbrace>"
affine_scaling: fixes s :: "'c set"   and c :: "real" assumes "affine s" shows "affine ((*\<^sub>R) c ` s)"
saturate_def: fixes f :: "'b \<Rightarrow> 'b" shows "saturate f = while (\<lambda>S. f S \<noteq> S) f"
prems_3: shows "dropWhile ((\<noteq>) v) (p # ps) = c"
meet_right: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L" shows "x \<sqinter> y \<sqsubseteq> y"
ListReds1: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and es :: "char list exp list" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>e # es,s,b\<rangle> [\<rightarrow>]* \<langle>e' # es,s',b'\<rangle>"
repeat_mset_iterate_add: fixes n :: "nat"   and M :: "'b multiset" shows "repeat_mset n M = iterate_add n M"
slice_take: fixes n :: "nat"   and w :: "'b word" shows "slice n w = of_bl (take (LENGTH('b) - n) (to_bl w))"
FVar: fixes accC :: "qtname"   and statDeclC :: "qtname"   and stat :: "bool"   and a :: "val"   and fn :: "vname" assumes "v = {accC,statDeclC,stat}Lit a..fn" shows "thesis"
eval_step_mmuaux_eq: shows "eval_step_mmuaux (tp, tss, len, maskL, maskR, a1_map, a2_map, done, done_length) = (tp, tl_queue tss', len - 1, maskL, maskR, a1_map, a2_map'', T # done, done_length + 1)"
bind_is_OK_pure_I: fixes f :: "('a, 'b, 'c) prog"   and h :: "'a"   and g :: "'c \<Rightarrow> ('a, 'b, 'd) prog" assumes "pure f h"   and "h \<turnstile> ok f"   and "\<And>x. h \<turnstile> f \<rightarrow>\<^sub>r x \<Longrightarrow> h \<turnstile> ok g x" shows "h \<turnstile> ok (f \<bind> g)"
inv_on_left_moving_norm_Oc_Oc: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" assumes "inv_on_left_moving_norm (as, lm) (s, l, Oc # r) ires"   and "hd l = Oc"   and "l \<noteq> []" shows "inv_on_left_moving_norm (as, lm) (s, tl l, Oc # Oc # r) ires"
class_add_subtype: fixes P :: "jvm_prog"   and x :: "ty"   and y :: "ty"   and C :: "char list"   and cdec :: "jvm_method class" assumes "subtype P x y"   and "\<not> is_class P C" shows "subtype (class_add P (C, cdec)) x y"
delta: fixes a :: "'b"   and g :: "'b \<Rightarrow> 'a" shows "(\<Prod>b. if b = a then g b else (1::'a)) = g a"
Gromov_extension_inside_space: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "Gromov_extension f (to_Gromov_completion x) = to_Gromov_completion (f x)"
derive_drop_prefixword: fixes w :: "symbol list"   and a :: "symbol list"   and b :: "symbol list" assumes "is_word w"   and "derives (w @ a) b" shows "derives a (drop (length w) b)"
assms: shows "((N, e), d) \<in> set_spmf I"
real_cond_exp_int_1: fixes f :: "'a \<Rightarrow> real" assumes "integrable M f" shows "integrable M (real_cond_exp M G f)"
intro: fixes hom :: "'c \<Rightarrow> 'd" assumes "field_hom hom" shows "field_hom' hom"
infsetsum'_Un_disjoint: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'c set" assumes "infsetsum'_converges f A"   and "infsetsum'_converges f B"   and "A \<inter> B = {}" shows "infsetsum' f (A \<union> B) = infsetsum' f A + infsetsum' f B"
h2: fixes u :: "nat \<Rightarrow> state" assumes "vs (first u) = c" shows "F (first s ## u)"
listsum_dims: fixes As :: "'a tensor list"   and ds :: "nat list" assumes "\<And>A. A \<in> set As \<Longrightarrow> dims A = ds" shows "dims (listsum ds As) = ds"
eq_iff: fixes x :: "'b"   and y :: "'b" shows "(x = y) = (c1.le x y \<and> c1.le y x)"
ih: assumes "valid ((x, T1) # \<Gamma>2)"   and "(x, T1) # \<Gamma>1 \<subseteq> (x, T1) # \<Gamma>2" shows "(x, T1) # \<Gamma>2 \<turnstile> t : T2"
polys_to_mat_def: fixes ts :: "'b list"   and ps :: "('b \<Rightarrow>\<^sub>0 'a) list" shows "polys_to_mat ts ps = mat_of_rows (length ts) (map (poly_to_row ts) ps)"
splus_mono_strict: fixes v :: "'t"   and w :: "'t"   and t :: "'a" assumes "v \<prec>\<^sub>t w" shows "t \<oplus> v \<prec>\<^sub>t t \<oplus> w"
all_partitions_alg_def: fixes X :: "'b set" shows "all_partitions_alg X = all_partitions_list (sorted_list_of_set X)"
transaction_strand_memberD: fixes x :: "'e strand_label \<times> (('f, 'g, 'h) prot_fun, (('f, 'g, 'h) prot_fun, 'g prot_atom) Term.term \<times> nat) stateful_strand_step"   and T :: "('f, 'g, 'h, 'e) prot_transaction" assumes "x \<in> set (transaction_strand T)" shows "x \<in> set (transaction_receive T) \<or> x \<in> set (transaction_selects T) \<or> x \<in> set (transaction_checks T) \<or> x \<in> set (transaction_updates T) \<or> x \<in> set (transaction_send T)"
mayDiverge_Par_R: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and c2 :: "('a, 'c) com"   and s :: "'b"   and c1 :: "('a, 'c) com" assumes "PL.mayDiverge tval aval c2 s" shows "PL.mayDiverge tval aval (c1 | c2) s"
hermitian_proj_fst_k: fixes k :: "nat" shows "adjoint (proj_fst_k k) = proj_fst_k k"
lossless_exec_gpv: fixes gpv :: "('c, 'b, bool list) gpv"   and s :: "'b \<Rightarrow> bool list option" assumes "lossless_gpv \<I>_full gpv"   and "\<And>s out. \<lbrakk>out \<in> outs_\<I> \<I>_full; finite (dom s)\<rbrakk> \<Longrightarrow> lossless_spmf (prf.random_oracle s out)"   and "\<I>_full \<turnstile>g gpv \<surd>"   and "finite (dom s)" shows "lossless_spmf (exec_gpv prf.random_oracle gpv s)"
S5: shows "(reflexive (r) \<and> universal (\<lambda>x y. \<lfloor>(\<lambda>z. x r y \<and> y r z \<longrightarrow> x r z)\<rfloor>)) \<and> universal (\<lambda>x y. x r y \<longrightarrow> y r x)"
awalkE: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c" assumes "wf_digraph G"   and "pre_digraph.awalk G u p v" obtains "set (pre_digraph.awalk_verts G u p) \<subseteq> verts G" and "set p \<subseteq> arcs G" and "pre_digraph.cas G u p v" and "hd (pre_digraph.awalk_verts G u p) = u" and "last (pre_digraph.awalk_verts G u p) = v"
CGA: shows "SIDE_GEN_ALGO (eq_linorder cmpk')"
rel_frontier_of_polyhedron: fixes S :: "'a set" assumes "polyhedron S" shows "rel_frontier S = \<Union> {F. F facet_of S}"
bounded_clinear_CBlinfun_apply: fixes f :: "'a \<Rightarrow> 'b" assumes "bounded_clinear f" shows "(*\<^sub>V) (CBlinfun f) = f"
clock_trans_def: fixes agents :: "'c odlist"   and jkbp :: "'c \<Rightarrow> ('c, 'd, 'e) GC list"   and envAction :: "'b \<Rightarrow> 'f list"   and envTrans :: "'f \<Rightarrow> ('c \<Rightarrow> 'e) \<Rightarrow> 'b \<Rightarrow> 'b"   and envVal :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and envObs :: "'c \<Rightarrow> 'b \<Rightarrow> 'g" shows "ClockView.clock_trans agents jkbp envAction envTrans envVal envObs \<equiv> \<lambda>cec X. fromList (concat (map (\<lambda>s. concat (map (\<lambda>eact. map (\<lambda>aact. envTrans eact aact s) (listToFuns (\<lambda>a. ClockView.clock_simAction jkbp envVal envObs a (cec, clock_knowledge envObs cec a s)) (toList agents))) (envAction s))) (toList X)))"
Nil_in_Suffixes_iff: fixes A :: "'a list set" shows "([] \<in> Suffixes A) = (A \<noteq> {})"
f8: shows "\<not> (\<exists>(a', b', c')\<in>set c. (a' = 0 \<and> b' \<noteq> 0) \<and> (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0))"
r_r_into_trancl: fixes a :: "'a"   and b :: "'a"   and R :: "('a \<times> 'a) set"   and c :: "'a" assumes "(a, b) \<in> R"   and "(b, c) \<in> R" shows "(a, c) \<in> R\<^sup>+"
is_extension: shows "\<forall>f. C.arr f \<longrightarrow> E.map (\<F>C.inclusion_of_generators f) = V f"
inside_outside: fixes S :: "'a set" shows "inside S = - (S \<union> outside S)"
modif_clause: shows "\<forall>\<sigma>. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/UNIV \<^esub>{\<sigma>} Call q (Modif \<sigma>),{}"
mono_onD: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and r :: "'b"   and s :: "'b" assumes "mono_on f A"   and "r \<in> A"   and "s \<in> A"   and "r \<le> s" shows "f r \<le> f s"
fup2: fixes f :: "'c \<rightarrow> 'b"   and x :: "'c" shows "fup\<cdot>f\<cdot>(up\<cdot>x) = f\<cdot>x"
simps: fixes a1 :: "'b measure"   and a2 :: "'b measure" shows "(a1 \<le> a2) = ((\<exists>M N. a1 = M \<and> a2 = N \<and> space M \<subset> space N) \<or> (\<exists>M N. a1 = M \<and> a2 = N \<and> space M = space N \<and> sets M \<subset> sets N) \<or> (\<exists>M N. a1 = M \<and> a2 = N \<and> space M = space N \<and> sets M = sets N \<and> emeasure M \<le> emeasure N))"
supr_least_Above: fixes r :: "('a \<times> 'a) set"   and B :: "'a set"   and a :: "'a" assumes "wo_rel r"   and "B \<subseteq> Field r"   and "a \<in> Above r B" shows "(wo_rel.supr r B, a) \<in> r"
div_mult_mult1_if: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "divide (times c a) (times c b) = (if c = zero then zero else divide a b)"
c_insert_df: shows "c_insert = (\<lambda>x u. set_to_nat (nat_to_set u \<union> {x}))"
eigen_vector_z_sr_c: shows "eigen_vector cA (map_vector complex_of_real z) (complex_of_real sr)"
assms: shows "u \<notin> G.scc_of (tail G a)"
src_vcomp: fixes \<nu> :: "'b"   and \<mu> :: "'b" assumes "D.seq \<nu> \<mu>" shows "src\<^sub>D (\<nu> \<cdot>\<^sub>D \<mu>) = src\<^sub>D \<nu>"
aeq: shows "subgroup_generated (reduced_homology_group (int p) (nsphere p)) {a} = reduced_homology_group (int p) (nsphere p)"
prod_cases5: fixes y :: "'c \<times> 'd \<times> 'e \<times> 'f \<times> 'g" obtains a :: "'c" and b :: "'d" and c :: "'e" and d :: "'f" and e :: "'g" where "y = (a, b, c, d, e)"
lappend_lprefixE: fixes xs :: "'b llist"   and ys :: "'b llist"   and zs :: "'b llist" assumes "lprefix (lappend xs ys) zs" obtains zs' :: "'b llist" where "zs = lappend xs zs'"
get_parent_locs_impl: shows "get_parent_locs = local.a_get_parent_locs"
worecSL_isSucc: fixes L :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c"   and i :: "'a"   and S :: "'a \<Rightarrow> 'c \<Rightarrow> 'c" assumes "adm_woL L"   and "isSucc i" shows "worecSL S L i = S (pred i) (worecSL S L (pred i))"
one_sup_induct: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<And>i. P (f i) (g i)"   and "\<And>j I. \<lbrakk>j \<notin> I; P (sup_monoid.sum f I) (sup_monoid.sum g I)\<rbrakk> \<Longrightarrow> P (f j \<squnion> sup_monoid.sum f I) (g j \<squnion> sup_monoid.sum g I)" shows "P (sup_monoid.sum f {k. True}) (sup_monoid.sum g {k. True})"
comp_cod_arr: fixes f :: "'c \<times> 'c"   and b :: "'c \<times> 'c" assumes "CopxC.arr f"   and "CopxC.cod f = b" shows "b \<odot> f = f"
suma_middle_comm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p"   and G :: "'p"   and H :: "'p"   and I :: "'p" assumes "A B C D E F SumA G H I" shows "A B C F E D SumA G H I"
commutes_with_composition: fixes x :: "'a"   and y :: "'a" assumes "x \<in> R"   and "y \<in> R" shows "\<eta> (x + y) = \<eta> x +' \<eta> y"
CptnModCondT: fixes P0 :: "'b com"   and s :: "'b"   and ys :: "('b com option \<times> 'b) list"   and b :: "'b set"   and P1 :: "'b com" assumes "(Some P0, s) # ys \<in> cptn_mod"   and "s \<in> b" shows "(Some (Cond b P0 P1), s) # (Some P0, s) # ys \<in> cptn_mod"
lcs_minus_1: fixes a :: "'a"   and x :: "'a"   and b :: "'a"   and y :: "'a" assumes "a + x = b + y" shows "lcs x y - x = a - gcs a b"
map_add_dom_app_simps_1: fixes m :: "'d"   and l2 :: "'d \<Rightarrow> 'e option"   and l1 :: "'d \<Rightarrow> 'e option" assumes "m \<in> dom l2" shows "(l1 ++ l2) m = l2 m"
inject: fixes x :: "'a list"   and ya :: "'a list" shows "(Array x = Array ya) = (x = ya)"
quietTransition: fixes P :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and Rs :: "('a, 'b, 'c) residual" assumes "quiet P"   and "\<Psi> \<rhd> P \<longmapsto> Rs" obtains P' :: "('a, 'b, 'c) psi" where "Rs = \<tau> \<prec> P'" and "quiet P'"
BNTr11: fixes f :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "S_inductive_set D"   and "f \<in> carrier D \<rightarrow> carrier D"   and "a \<in> carrier D"   and "\<forall>x\<in>carrier D. x \<preceq> f x" shows "f (Algebra2.Sup D (\<Union> (WWa D f a))) = Algebra2.Sup D (\<Union> (WWa D f a))"
eventually_at_left_field: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c" shows "eventually P (at_left x) = (\<exists>b<x. \<forall>y>b. y < x \<longrightarrow> P y)"
steps_Throw_impl_exec: fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and c :: "('a, 'b, 'c) com"   and s :: "('a, 'c) xstate"   and t :: "'a" assumes "\<Gamma>\<turnstile> (c, s) \<rightarrow>\<^sup>* (Throw, Normal t)" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> Abrupt t"
logical_instr_def: fixes instr :: "sparc_operation \<times> inst_operand list" shows "logical_instr instr \<equiv> let instr_name = fst instr; op_list = snd instr; rs1 = get_operand_w5 (op_list ! 1); rd = get_operand_w5 (op_list ! 3) in do operand2 \<leftarrow> gets (get_operand2 op_list); curr_win \<leftarrow> get_curr_win (); rs1_val \<leftarrow> gets (user_reg_val curr_win rs1); rd_val \<leftarrow> gets (user_reg_val curr_win rd); result \<leftarrow> gets (\<lambda>s. logical_result instr_name rs1_val operand2); new_rd_val \<leftarrow> gets (\<lambda>s. if rd \<noteq> 0 then result else rd_val); write_reg new_rd_val curr_win rd; logical_instr_sub1 instr_name result od"
reduced_GB_ideal_Polys: fixes X :: "'a option set"   and F :: "(('a option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set" assumes "finite X"   and "F \<subseteq> P[X]" shows "ideal (extended_ord.punit.reduced_GB F) = ideal F"
rename_LeadsTo: fixes h :: "'c \<Rightarrow> 'd"   and F :: "'c program"   and A :: "'c set"   and B :: "'c set" assumes "bij h" shows "(rename h F \<in> h ` A \<longmapsto>w h ` B) = (F \<in> A \<longmapsto>w B)"
Macaulay_red_alt: fixes fs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "Macaulay_red (Keys_to_list fs) fs = filter (\<lambda>p. lt p \<notin> lt_set (set fs)) (Macaulay_list fs)"
that: fixes k :: "'a \<Rightarrow> 'b" assumes "k \<in> carrier (DirProds G I)"   and "k i = x" shows "thesis"
HNatInfinite_upward_closed: fixes x :: "nat star"   and y :: "nat star" assumes "x \<in> HNatInfinite"   and "x \<le> y" shows "y \<in> HNatInfinite"
assms_3: shows "t' \<in> existence_ivl0 y"
rel_map_2: fixes Sa :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a pmf"   and g :: "'d \<Rightarrow> 'b"   and y :: "'d pmf" shows "rel_pmf Sa x (map_pmf g y) = rel_pmf (\<lambda>x y. Sa x (g y)) x y"
simps_2: fixes n :: "nat" shows "lang n Full = lists (\<Sigma> n)"
mult_cancel_right2: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a" assumes "class.ring_1_no_zero_divisors one times plus zero minus uminus" shows "(times a c = c) = (c = zero \<or> a = one)"
ccSUP_mono: fixes A :: "'c set"   and B :: "'d set"   and f :: "'c \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'a" assumes "countable A"   and "countable B"   and "\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. f n \<le> g m" shows "Sup (f ` A) \<le> Sup (g ` B)"
lift_def: fixes f :: "'a \<Rightarrow> 'b" shows "lift f = return \<circ> f"
integral_sum: fixes I :: "'e set"   and M :: "'f measure"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'g" assumes "\<And>i. i \<in> I \<Longrightarrow> integrable M (f i)" shows "\<integral> x. (\<Sum>i\<in>I. f i x) \<partial>M = (\<Sum>i\<in>I. integral\<^sup>L M (f i))"
assms_4: fixes P :: "'d"   and Q :: "'d" assumes "Prop P Q" shows "\<Psi> \<rhd> C P \<leadsto>\<guillemotleft>weakBisim\<guillemotright> C Q"
knC_resp: fixes runz :: "fid_t \<Rightarrow> (role_t \<times> agent list \<times> atom list) option"   and Rb :: "fid_t"   and A :: "agent"   and B :: "agent"   and K :: "key"   and al :: "atom list" assumes "runz Rb = Some (Resp, [A, B], aKey K # al)" shows "(K, B) \<in> knC runz"
L3: shows "{c_fst z |z. z \<in> A} \<in> ce_sets"
two_t_designs_eq_points_axioms: shows "two_t_designs_eq_points \<V> \<B> \<k> \<B>' 2 \<Lambda> \<Lambda>'"
b_e_type_block: fixes \<C> :: "t_context"   and e :: "b_e"   and ts :: "t list"   and ts' :: "t list"   and tf :: "tf"   and es :: "b_e list" assumes "\<C> \<turnstile> [e] : ts _> ts'"   and "e = Block tf es" shows "\<exists>ts'' tfn tfm. tf = tfn _> tfm \<and> ts = ts'' @ tfn \<and> ts' = ts'' @ tfm \<and> \<C>\<lparr>label := [tfm] @ label \<C>\<rparr> \<turnstile> es : tf"
lOfWax_def: fixes \<sigma> :: "'c" shows "lOfWax \<sigma> = Fn (Tag \<sigma>) [rOfWax \<sigma>]"
search_enum_empty: fixes b :: "nat"   and i :: "nat"   and x :: "nat" assumes "s b \<down>= i"   and "\<And>x. \<phi> i x \<up>" shows "search_enum x b \<up>"
select_convs_5: fixes nothing :: "unit"   and nothinga :: "unit"   and doctype :: "char list"   and document_element :: "(_) element_ptr option"   and disconnected_nodes :: "(_) node_ptr list"   and more :: "'m" shows "RDocument.more \<lparr>RObject.nothing = nothing, RDocument.nothing = nothinga, doctype = doctype, document_element = document_element, disconnected_nodes = disconnected_nodes, \<dots> = more\<rparr> = more"
rot_circle_eq_circle: fixes x :: "real"   and y :: "real" assumes "0 \<le> x"   and "x \<le> 1"   and "0 \<le> y"   and "y \<le> 1" shows "rot_circle_cube (x, y) = circle_cube (x_coord_inv ((2 * x - 1) * circle_y (x_coord y)), x_coord y / (2 * circle_y (x_coord (x_coord_inv ((2 * x - 1) * circle_y (x_coord y))))) + 1 / 2)"
conv_radius_mult_ge: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" shows "min (conv_radius f) (conv_radius g) \<le> conv_radius (\<lambda>x. \<Sum>i\<le>x. f i * g (x - i))"
verts_app_inv_iso_subgraph: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism"   and V :: "'a set" assumes "digraph_isomorphism hom"   and "V \<subseteq> verts G" shows "iso_verts (inv_iso hom) ` iso_verts hom ` V = V"
IH: fixes S :: "nat set" assumes "\<forall>a\<in>S. length xs \<le> a" shows "nths (xs @ ys) S = nths ys {x - length xs |x. x \<in> S}"
init_segment_trans: fixes S :: "nat set"   and T :: "nat set"   and U :: "nat set" assumes "init_segment S T"   and "init_segment T U" shows "init_segment S U"
S2_successors: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> $S2 rmhist p \<and> ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<longrightarrow> S2 rmhist p$ \<or> S3 rmhist p$"
N1: fixes A :: "nat set" assumes "A \<in> [N1]\<^bsup>4\<^esup>" shows "f1 A = j1"
zero_not_one: shows "(0::'b) \<noteq> (1::'b)"
sup_eq_neg_inf: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "sup a b = uminus (inf (uminus a) (uminus b))"
WT_resource_of_oracle: fixes \<I> :: "('b, 'c) \<I>"   and "oracle" :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'a) spmf"   and s :: "'a" assumes "\<And>s. \<I> \<turnstile>c oracle s \<surd>" shows "\<I> \<turnstile>res RES oracle s \<surd>"
mcont_bind_spmf: fixes luba :: "'a set \<Rightarrow> 'a"   and orda :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b spmf"   and g :: "'b \<Rightarrow> 'a \<Rightarrow> 'c spmf" assumes "mcont luba orda lub_spmf (ord_spmf (=)) f"   and "\<And>y. mcont luba orda lub_spmf (ord_spmf (=)) (g y)" shows "mcont luba orda lub_spmf (ord_spmf (=)) (\<lambda>x. f x \<bind> (\<lambda>y. g y x))"
V_axiom_on_def: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" shows "V_axiom_on A f = (\<forall>B\<subseteq>A. \<forall>y\<in>B. (\<forall>x\<in>B. (x, y) \<in> rwp_on A f) \<longrightarrow> y \<in> f B)"
awalk_verts_arc2: fixes G :: "('b, 'c) pre_digraph"   and u :: "'b"   and p :: "'c list"   and v :: "'b"   and e :: "'c" assumes "wf_digraph G"   and "pre_digraph.awalk G u p v"   and "e \<in> set p" shows "head G e \<in> set (pre_digraph.awalk_verts G u p)"
exp: shows "is_expansion (MS (MSLCons (C, e) xs) f') basis"
assms_1: shows "InvariantVarsM M F0 Vbl"
Range_empty_iff: fixes r :: "('c \<times> 'b) set" shows "(Range r = {}) = (r = {})"
assms_1: fixes s :: "(name \<times> name) list" shows "P[
] \<leadsto>\<guillemotleft>weakBisim\<guillemotright> Q[
]"
x_zero_interr: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> (0::'a) \<parallel> (y \<cdot> (0::'a)) = x \<parallel> y \<cdot> (0::'a)"
comp_cod_arr: fixes f :: "('o, 'a) arr"   and b :: "('o, 'a) arr" assumes "arr f"   and "cod f = b" shows "COMP b f = f"
TS_OPT2': fixes x :: "nat"   and y :: "nat"   and \<sigma> :: "nat list" assumes "x \<noteq> y"   and "set \<sigma> \<subseteq> {x, y}" shows "real (T\<^sub>p_on (rTS []) [x, y] \<sigma>) \<le> 2 * real (T\<^sub>p_opt [x, y] \<sigma>) + 2"
lossless_cond_spmf_fst: fixes p :: "('b \<times> 'a) spmf"   and x :: "'b" shows "lossless_spmf (cond_spmf_fst p x) = (x \<in> fst ` set_spmf p)"
elims_2: fixes x :: "nat"   and xa :: "nat ifex"   and xb :: "bdd" assumes "Rmi_g x xa xb" obtains bdd :: "bdd" where "x = 0" and "xa = Falseif" and "xb = bdd__"   | bdd :: "bdd" where "x = Suc 0" and "xa = Trueif" and "xb = bdda__"   | n :: "nat" and v :: "nat" and t :: "nat ifex" and e :: "nat ifex" and bdd :: "bdd" where "x = Suc (Suc n)" and "xa = IF v t e" and "xb = bdd" and "pointermap_p_valid n (dpm bdd) \<and> (case pm_pth (dpm bdd) n of (nv, nt, ne) \<Rightarrow> nv = v \<and> Rmi_g nt t bdd \<and> Rmi_g ne e bdd)"
id: shows "concat (map (\<lambda>i. if v $ i \<noteq> (0::'a) then [i] else []) [0..<n]) = k # ids"
Quotient3_to_Quotient_equivp: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b"   and T :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "Quotient3 R Abs Rep"   and "T \<equiv> \<lambda>x. (=) (Abs x)"   and "equivp R" shows "Quotient R Abs Rep T"
Pred: fixes P :: "'state"   and \<phi> :: "'pred" assumes "P \<turnstile> \<phi>" shows "P \<rightarrow>\<^sub>S \<langle>S_action.Pred \<phi>,P\<rangle>"