hcju/isabelleps huggingface dataset:
rep_eq: fixes x :: "'f \<Rightarrow> bool"   and xa :: "('f, 'e) generator" shows "generator (dropWhile_trans x xa) = dropWhile_raw x (generator xa)"
numneg: fixes bs :: "real list"   and t :: "MIR.num" shows "Inum bs (numneg t) = Inum bs (Neg t)"
induct: fixes P :: "'a list \<Rightarrow> bool"   and a0 :: "'a list" assumes "P []"   and "\<And>y ys. (\<And>x xa. \<lbrakk>x = fold min_for ys y; xa = remove1 x (y # ys)\<rbrakk> \<Longrightarrow> P xa) \<Longrightarrow> P (y # ys)" shows "P a0"
biimp_def: fixes F :: "'b formula"   and G :: "'b formula" shows "F \<^bold>\<leftrightarrow> G \<equiv> (F \<^bold>\<rightarrow> G) \<^bold>\<and> (G \<^bold>\<rightarrow> F)"
Sup_least: 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 A :: "'a set"   and z :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>x. s2p A x \<Longrightarrow> less_eq x z" shows "less_eq (Sup A) z"
NextRecvUpd'_preserves_prop_safe: fixes c0 :: "('b, 't, 'loc) Combined.configuration"   and q :: "'b"   and c1 :: "('b, 't, 'loc) Combined.configuration"   and p :: "'b" assumes "prop_safe (prop_config c0 q)"   and "InvGlobPointstampsEq c0"   and "cri.InvMsgInGlob (exchange_config c0)"   and "NextRecvUpd' c0 c1 p q" shows "prop_safe (prop_config c1 q)"
mult_pos_neg: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less zero a"   and "less b zero" shows "less (times a b) zero"
nchotomy: shows "\<forall>accent. accent = Acute \<or> accent = Grave \<or> accent = Macron"
cases: fixes x :: "('e, 'd) Term.term" obtains xa :: "'d" where "x = Var xa"   | f :: "'e" and ts :: "('e, 'd) Term.term list" where "x = Fun f ts"
Leaf_in_set_random_bst_iff: fixes A :: "'a set" shows "(\<langle>\<rangle> \<in> set_pmf (random_bst A)) = (A = {} \<or> infinite A)"
pure_def: fixes R :: "('d \<times> 'c) set" shows "pure R \<equiv> \<lambda>a c. \<up> ((c, a) \<in> R)"
finite_reduce: fixes f :: "nat \<Rightarrow> real" shows "finite (reduce f PR)"
join_connection2: fixes x :: "'a"   and y :: "'a" shows "(x \<sqsubseteq> y) = (x \<squnion> y = y)"
empty_filter_conv: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "([] = filter P xs) = (\<forall>x\<in>set xs. \<not> P x)"
assms_3: shows "w0 \<in> \<langle>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}\<rangle>"
right_quotient_def: fixes u :: "'b list"   and v :: "'b list" shows "u\<^sup><\<inverse>v  = rev (rev v\<inverse>\<^sup>>rev u)"
While': fixes b :: "bexp"   and l :: "nat"   and c :: "com" assumes "sec b \<le> l"   and "l \<turnstile>' c" shows "l \<turnstile>' WHILE b DO c"
vdisjnt_self_iff_vempty: fixes S :: "V" shows "vdisjnt S S = (S = 0)"
assms_2: shows "\<forall>x xs. P (x # x # xs) = P (x # xs)"
assms_5: shows "M' = p @ [(bLiteral, False)]"
inf_sup_ord_4: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and y :: "'d"   and x :: "'d" assumes "class.lattice inf less_eq less sup" shows "less_eq y (sup x y)"
components_are_iso: fixes a :: "'a \<times> 'a \<times> 'a" assumes "B.VVV.ide a" shows "B.iso (B.HoVH a)"
less_infinityE: fixes n :: "eint" assumes "n < \<infinity>" obtains k :: "int" where "n = eint k"
simps: fixes I :: "interp"   and "is" :: "int list"   and i :: "int"   and d :: "int" shows "I \<Turnstile>0 presb.Eq is i d = (eval_tm I is = i - 2 ^ Length I * d)"
assms_5: shows "AE x in N. x \<in> \<Omega>"
refine: fixes n :: "nat"   and Xi :: "(real list \<times> real list) list"   and X :: "'b set"   and sctnsi :: "real list sctn list"   and sctns :: "'b sctn set" assumes "DIM_precond TYPE('b) n"   and "(Xi, X) \<in> clw_rel lvivl_rel"   and "(sctnsi, sctns) \<in> sctns_rel" shows "(nres_of (restrict_to_halfspaces_clw_rel n sctnsi Xi), restrict_to_halfspaces_clw sctns X) \<in> \<langle>clw_rel lvivl_rel\<rangle>nres_rel"
dom_char: fixes f :: "'a" shows "S.dom f = (if S.arr f then local.dom f else null)"
safety_prop_stable: fixes A :: "'c set" shows "safety_prop (stable A)"
addrow_mat_def: fixes n :: "nat"   and a :: "'a"   and k :: "nat"   and l :: "nat" shows "addrow_mat n a k l = mat n n (\<lambda>(i, j). (if k = i \<and> l = j then (+) a else id) (if i = j then 1::'a else (0::'a)))"
some_geodesic_compact: fixes x :: "'b"   and S :: "'b set"   and y :: "'b" shows "compact {x--S--y}"
enc: fixes var :: "'d set"   and trm :: "'e set"   and fmla :: "'f set"   and Var :: "'d \<Rightarrow> 'e"   and FvarsT :: "'e \<Rightarrow> 'd set"   and substT :: "'e \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> 'e"   and Fvars :: "'f \<Rightarrow> 'd set"   and subst :: "'f \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> 'f"   and num :: "'e set"   and enc :: "'f \<Rightarrow> 'e"   and \<phi> :: "'f" assumes "Encode var trm fmla Var FvarsT substT Fvars subst num enc"   and "\<phi> \<in> fmla" shows "enc \<phi> \<in> num"
orderE: fixes b :: "'k multiset"   and a :: "'k multiset" assumes "b \<subseteq># a" obtains "a = a \<union># b"
vars_term_subst: fixes t :: "('d, 'e) Term.term"   and \<sigma> :: "'e \<Rightarrow> ('d, 'c) Term.term" shows "vars_term (t \<cdot> \<sigma>) = \<Union> (vars_term ` \<sigma> ` vars_term t)"
stable_on_range': fixes ds :: "'b set"   and X :: "'a set"   and x :: "'a" assumes "Pdds'.stable_on ds X"   and "x \<in> X" shows "x \<in> Field (Pd_shuffle_to_top (insert d ds') Y (Xd x))"
bij_inv_1: fixes x :: "'a" assumes "is_bij x" shows "x ; x\<^sup>\<smile> = 1'"
valid_write_sops_nth_update: fixes i :: "nat"   and sb :: "'a memref list"   and p :: "'a"   and "is" :: "instr list"   and xs :: "nat \<Rightarrow> nat option"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "\<forall>sop\<in>write_sops sb. valid_sop sop" shows "valid_write_sops (ts[i := (p, is, xs, sb, \<D>, \<O>, \<R>)])"
increasing3: fixes ls :: "'a list" assumes "\<forall>as bs. ls = as @ bs \<longrightarrow> (\<forall>x\<in>set as. \<forall>y\<in>set bs. x \<le> y)" shows "increasing ls"
esl_def: 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" shows "SemiType.esl P \<equiv> (types P, subtype P, SemiType.sup P)"
prems_1: fixes a :: "'c" assumes "a \<in> insert x F" shows "f a \<otimes> g a = h a"
igSubstIGVar2_fromMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "gSubstGVar2 MOD" shows "igSubstIGVar2 (fromMOD MOD)"
F: shows "(F, F') \<in> (Rb \<rightarrow> Ra) \<rightarrow> Rb \<rightarrow> Ra"
Bonk_Schramm_extension_unfolded_wo_props'_2: fixes b :: "'a Bonk_Schramm_extension_unfolded"   and a :: "'a Bonk_Schramm_extension_unfolded" shows "b \<in> underS Bonk_Schramm_extension_unfolded_wo (middle a b)"
children_h': shows "h' \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r insert_before_list node reference_child children_h3"
lspace_ennreal_iff: fixes p :: "real"   and S :: "'a set" assumes "0 < p" shows "lspace (lebesgue_on S) (ennreal p) = {f \<in> borel_measurable (lebesgue_on S). integrable (lebesgue_on S) (\<lambda>x. norm (f x) powr p)}"
insertI: fixes x :: "'e"   and A :: "'e set"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'f"   and z :: "'f"   and y :: "'f" assumes "x \<notin> A"   and "fold_graph f z A y" shows "fold_graph f z (insert x A) (f x y)"
discI: fixes list :: "'a list"   and x21 :: "'a"   and x22 :: "'a list" assumes "list = x21 \<Colon> x22" shows "list \<noteq> []"
callee: fixes x :: "'a"   and s :: "'s" assumes "x \<in> outs_\<I> \<I>"   and "I s" shows "lossless_spmf (callee s x)"
dom_trg: fixes \<mu> :: "'b" assumes "D.arr \<mu>" shows "D.dom (trg\<^sub>D \<mu>) = trg\<^sub>D \<mu>"
maximum: shows "rdom \<lceil>\<lambda>s. True\<rceil> \<subseteq> wp (IF (\<lambda>s. s ''y'' \<le> s ''x'') THEN (''z'' ::= (\<lambda>s. s ''x'')) ELSE (''z'' ::= (\<lambda>s. s ''y'')) FI) \<lceil>\<lambda>s. s ''z'' = max (s ''x'') (s ''y'')\<rceil>"
assms_3: fixes i :: "nat" assumes "i < n" shows "v $ i < w $ i"
char_mult: fixes a :: "nat"   and b :: "nat" assumes "a \<in> carrier G"   and "b \<in> carrier G" shows "\<chi> (a \<otimes>\<^bsub>G\<^esub> b) = \<chi> a * \<chi> b"
derivation_shift_identity: fixes D :: "(nat \<times> symbol \<times> symbol list) list" shows "derivation_shift D 0 0 = D"
the_NF: fixes A :: "('b \<times> 'b) set"   and a :: "'b" assumes "SN A"   and "CR A" shows "(a, the_NF A a) \<in> A\<^sup>!"
Predict_elem_in_Gen: 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 I :: "item set"   and Doc :: "character list"   and P :: "(symbol \<times> character list) list set"   and k :: "nat"   and x :: "item" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel"   and "I \<subseteq> LocalLexing.Gen \<NN> \<TT> \<RR> \<SS> Doc P"   and "k \<le> length Doc"   and "x \<in> LocalLexing.Predict \<RR> k I" shows "x \<in> LocalLexing.Gen \<NN> \<TT> \<RR> \<SS> Doc P"
compl_bot_eq: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "uminus bot = top"
diff_same: shows "gt_diff s' s \<or> gt_same s' s"
fold_conv_fold_keys: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and rbt :: "'c set_rbt"   and b :: "'b" shows "RBT_Set2.fold f rbt b = fold f (RBT_Set2.keys rbt) b"
cond2: shows "inv_u * Q $$ (1, 1) * (Matrix.mat 1 2 (\<lambda>(i, j). if i = 0 \<and> j = 0 then a else b) * Q) $$ (0, 0) = a"
rmra: shows "generalized_sfw (lr_of_tran_s1 rt) p = Some (rm, ra) \<and> ra = output_iface (routing_table_semantics rt (p_dst p))"
lfinite_lrel_eq: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c llist"   and ys :: "'d llist" assumes "lrel R xs ys" shows "lfinite xs = lfinite ys"
cond_D_def: fixes x :: "nat list"   and y :: "nat list" shows "cond_D x y = (\<forall>l\<le>n. take l b \<bullet> take l y \<le> a \<bullet> x)"
lens_indep_def: fixes X :: "'b \<Longrightarrow> 'c"   and Y :: "'d \<Longrightarrow> 'c" shows "X \<bowtie> Y \<equiv> (\<forall>\<sigma> v u. put\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> \<sigma> v) u = put\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> \<sigma> u) v) \<and> (\<forall>\<sigma> v. get\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> \<sigma> v) = get\<^bsub>X\<^esub> \<sigma>) \<and> (\<forall>\<sigma> u. get\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> \<sigma> u) = get\<^bsub>Y\<^esub> \<sigma>)"
scaleC_adj: fixes a :: "complex"   and A :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a" shows "(a *\<^sub>C A)* = cnj a *\<^sub>C A*"
well_related_iff_neg_well_founded: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "well_related_set A less_eq = well_founded A (\<lambda>x y. \<not> less_eq y x)"
forall_from_nat_rw: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b"   and P :: "'b \<Rightarrow> bool" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "(\<forall>x\<in>{0..<CARD('b)}. P (mod_type.from_nat Abs x)) = (\<forall>x. P (mod_type.from_nat Abs x))"
equivp: fixes cmp :: "'b \<Rightarrow> 'b \<Rightarrow> comp" assumes "comparator cmp" shows "equivp (\<lambda>a b. cmp a b = Equiv)"
list_assn_monot: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and P' :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and l :: "'a list"   and l' :: "'b list" assumes "\<And>x x'. P x x' \<Longrightarrow>\<^sub>t P' x x'" shows "list_assn P l l' \<Longrightarrow>\<^sub>t list_assn P' l l'"
sdrop_shift: fixes i :: "nat"   and w :: "'a list"   and s :: "'a stream" shows "sdrop i (w @- s) = drop i w @- sdrop (i - length w) s"
less_def: fixes a :: "'a"   and b :: "'a" shows "(a < b) = (a \<le> b \<and> \<not> b \<le> a)"
validInputTransition: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" assumes "P \<longmapsto> a<x> \<prec> P'"   and "valid P" shows "valid P'"
S_abss_\<S>: fixes s :: "'s \<times> ('c \<Rightarrow> real)" assumes "s \<in> S" shows "abss s \<in> \<S>"
star_n_one_num: shows "1 = star_n (\<lambda>n. 1::'a)"
enum_all_finite_4_def: fixes P :: "finite_4 \<Rightarrow> bool" shows "enum_all P = (P a\<^sub>1 \<and> P a\<^sub>2 \<and> P a\<^sub>3 \<and> P a\<^sub>4)"
tMap_is_tStep_conv: fixes f :: "'b \<Rightarrow> 'a"   and t :: "'b Trace"   and tp :: "'a Trace"   and s :: "'a" shows "(tMap f t = tp \<leadsto> s) = (\<exists>tp' s'. t = tp' \<leadsto> s' \<and> s = f s' \<and> tMap f tp' = tp)"
that: fixes LL :: "'a multiset" assumes "LL \<in> set Ci"   and "L \<in># LL" shows "thesis"
assms_1: shows "\<exists>i>\<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>. \<parallel>c\<parallel>\<^bsub>t i\<^esub>"
in_arcsD1: fixes G :: "'b pair_pre_digraph"   and u :: "'b"   and v :: "'b" assumes "pair_wf_digraph G"   and "(u, v) \<in> parcs G" shows "u \<in> pverts G"
finite_mod_type: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "finite UNIV"
content_normalize: fixes p :: "'a poly" shows "content (normalize p) = content p"
ESem_Let: fixes as :: "(Vars.var \<times> exp) list"   and body :: "exp"   and \<rho> :: "Vars.var \<Rightarrow> Value" shows "\<lbrakk> Terms.Let as body \<rbrakk>\<^bsub>\<rho>\<^esub> = (\<Lambda> x. x)\<cdot>(\<lbrakk> body \<rbrakk>\<^bsub>\<lbrace>as\<rbrace>\<rho>\<^esub>)"
p_boundsI: fixes x :: "real"   and y :: "real" assumes "(\<Sum>i<k. as ! i * bs ! i powr x) \<le> 1 \<and> 1 \<le> (\<Sum>i<k. as ! i * bs ! i powr y)" shows "p \<in> {y..x}"
ereal_Sup: fixes A :: "real set" assumes "\<bar>Sup (ereal ` A)\<bar> \<noteq> \<infinity>" shows "ereal (Sup A) = Sup (ereal ` A)"
assms_1: fixes x :: "'a" assumes "x \<in> t" shows "(g has_derivative g' x) (at x within t)"
fps_to_fls_nonzeroI: fixes f :: "'b fps" assumes "f \<noteq> 0" shows "fps_to_fls f \<noteq> 0"
splitAt_distinct_fst_snd: fixes vs :: "'c list"   and ram :: "'c" assumes "distinct vs" shows "set (fst (splitAt ram vs)) \<inter> set (snd (splitAt ram vs)) = {}"
lt_ex: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.no_bot less_eq less" shows "\<exists>y. less y x"
prj_simps_2: shows "C.seq \<mu>\<nu>.prj\<^sub>0 \<p>\<^sub>1[\<nu>.leg0 \<cdot> \<mu>\<nu>.prj\<^sub>0, \<pi>\<rho>.leg1]"
T_intersec_meas_1: fixes A :: "'a set"   and B :: "'a set" assumes "A \<in> sets M"   and "B \<in> sets M" shows "A \<inter> T -` B \<in> sets M"
total_quasi_ordered_set_axioms: shows "total_quasi_ordered_set A (\<sqsubseteq>)"
polypoly'_def: fixes bs :: "'b list"   and p :: "'b poly" shows "polypoly' bs p = map (Ipoly bs \<circ> decrpoly) (coefficients p)"
trivial_group_subgroup_generated: fixes G :: "('e, 'f) monoid_scheme"   and S :: "'e set" assumes "Group.group G"   and "S \<subseteq> {\<one>\<^bsub>G\<^esub>}" shows "trivial_group (subgroup_generated G S)"
hom_sum: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'c" assumes "A \<subseteq> carrier V"   and "g \<in> A \<rightarrow> carrier V" shows "T (finsum V g A) = (\<Oplus>\<^bsub>W\<^esub>a\<in>A. T (g a))"
step_imp_seq: fixes a :: "'a"   and b :: "'a"   and ars :: "('a \<times> 'b \<times> 'a) set" assumes "(a, b) \<in> unlabel ars" shows "\<exists>ss\<in>seq ars. fst ss = a \<and> lst ss = b"
entailsD: fixes P :: "'b \<Rightarrow> real"   and Q :: "'b \<Rightarrow> real"   and s :: "'b" assumes "P \<tturnstile> Q" shows "P s \<le> Q s"
spr_simRels_def: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and envInit :: "('b, 'e, 'f) BEState list"   and envAction :: "('b, 'e, 'f) BEState \<Rightarrow> 'g list"   and envTrans :: "'g \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('b, 'e, 'f) BEState \<Rightarrow> ('b, 'e, 'f) BEState"   and envVal :: "('b, 'e, 'f) BEState \<Rightarrow> 'c \<Rightarrow> bool"   and agents :: "'b odlist"   and envObsC :: "'e \<Rightarrow> 'h" assumes "FiniteDetBroadcastEnvironment jkbp envInit envAction envTrans envVal agents envObsC" shows "FiniteDetBroadcastEnvironment.spr_simRels envObsC \<equiv> \<lambda>a. rel_ext (\<lambda>uu_. \<exists>s s'. uu_ = (s, s') \<and> (envObsC (es (sprFst s)), ODList.lookup (ps (sprFst s)) a) = (envObsC (es (sprFst s')), ODList.lookup (ps (sprFst s')) a) \<and> (envObsC (es (sprLst s)), ODList.lookup (ps (sprLst s)) a) = (envObsC (es (sprLst s')), ODList.lookup (ps (sprLst s')) a) \<and> sprCRel s = sprCRel s')"
l5: fixes x :: "'a"   and y :: "'a" assumes "- - x \<le> - - y" shows "- y \<le> - x"
assms_2: shows "\<forall>B\<subseteq>A. f (f B) = f B"
filterlim_tendsto_add_at_top: fixes f :: "'b \<Rightarrow> real"   and c :: "real"   and F :: "'b filter"   and g :: "'b \<Rightarrow> real" assumes "(f \<longlongrightarrow> c) F"   and "filterlim g at_top F" shows "LIM x F. f x + g x :> at_top"
T_def: fixes f :: "'a \<times> 'a" shows "T f \<equiv> if CC.arr f then fst f \<otimes> snd f else null"
simps_4: fixes \<xi> :: "nat \<Rightarrow> 'a set"   and q :: "'a" shows "\<xi> \<Turnstile>\<^sub>n nprop\<^sub>n(q) = (q \<notin> \<xi> 0)"
pfun_upd_twice: fixes f :: "('a, 'b) pfun"   and x :: "'a"   and u :: "'b"   and v :: "'b" shows "f(x \<mapsto> u, x \<mapsto> v)\<^sub>p = f(x \<mapsto> v)\<^sub>p"
setdist_closure_1: fixes S :: "'a set"   and T :: "'a set" shows "setdist (closure S) T = setdist S T"
dual_anti_iff: fixes x :: "'b"   and y :: "'b" shows "(x \<le> y) = (\<partial> y \<le> \<partial> x)"
primroot_ex: fixes x :: "'b list" assumes "x \<noteq> \<epsilon>" shows "\<exists>r k. x \<in>\<^sub>p r * \<and> k \<noteq> 0 \<and> x = r \<^sup>@ k"
distinct_disc_1: fixes pr_operation :: "pr_operation" assumes "is_PUSH pr_operation" shows "\<not> is_RELABEL pr_operation"
instantiate_var_same_type'': fixes a :: "Core.term"   and \<tau> :: "typ"   and B :: "Core.term"   and x :: "variable" assumes "typ_of a = Some \<tau>"   and "is_closed B" shows "Abs_fv x \<tau> B \<bullet> a = subst_term [((x, \<tau>), a)] B"
simps_4: fixes m :: "'node"   and m' :: "'node"   and S :: "'node SDG_node set"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and p :: "'pname"   and f :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option" shows "slice_kind_aux m m' S (Q\<hookleftarrow>\<^bsub>p\<^esub>f) = (if m \<in> \<lfloor>S\<rfloor>\<^bsub>CFG\<^esub> then let outs = THE outs. \<exists>ins. (p, ins, outs) \<in> set procs in Q\<hookleftarrow>\<^bsub>p\<^esub>\<lambda>cf cf'. rspp m' S outs cf' cf else \<lambda>cf. True\<hookleftarrow>\<^bsub>p\<^esub>\<lambda>cf cf'. cf')"
a1: shows "\<forall>s. Q' s \<longrightarrow> Q (s(v := e s))"
expand: fixes prod :: "'a \<times> 'b"   and prod' :: "'a \<times> 'b" assumes "fst prod = fst prod' \<and> snd prod = snd prod'" shows "prod = prod'"
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 :: "('c, 'a, 'b, 'e, 'g option) RElement_scheme \<Rightarrow> 'k" 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'"
invf_def: fixes K :: "('c, 'd) Ring_scheme"   and x :: "'c" shows "x\<^bsup>\<hyphen> K\<^esup> = (THE y. y \<in> carrier K \<and> y \<cdot>\<^sub>r\<^bsub>K\<^esub> x = 1\<^sub>r\<^bsub>K\<^esub>)"
R_def: shows "R = A - (\<Union>i. (T ^^ i) --` B)"
is2_valid: shows "is2 \<lhd> dims (tensors_from_net (witness_l (tl rs)) $ 0)"
\<phi>_in_terms_of_\<eta>: fixes y :: "'d"   and f :: "'c"   and x :: "'c" assumes "D.ide y"   and "\<guillemotleft>f : F y \<rightarrow>\<^sub>C x\<guillemotright>" shows "\<phi> y f = G f \<cdot>\<^sub>D \<eta> y"
set_mset_remdups_mset: fixes A :: "'b multiset" shows "set_mset (remdups_mset A) = set_mset A"
not_eq_order_implies_strict: fixes a :: "'d list"   and b :: "'d list" assumes "a \<noteq> b"   and "prefix a b" shows "strict_prefix a b"
conv_add: fixes x :: "'a"   and y :: "'a" shows "(x + y)\<^sup>T = x\<^sup>T + y\<^sup>T"
DIGeqaxiom_def: shows "DIGeqaxiom = (($\<phi> vid1 local.empty \<rightarrow> [[EvolveODE (OVar vid1) ($\<phi> vid1 local.empty)]]Geq (Differential (f1 fid1 vid1)) (Differential (f1 fid2 vid1))) \<rightarrow> ($\<phi> vid1 local.empty \<rightarrow> Geq (f1 fid1 vid1) (f1 fid2 vid1)) \<rightarrow> [[EvolveODE (OVar vid1) ($\<phi> vid1 local.empty)]]Geq (f1 fid1 vid1) (f1 fid2 vid1))"
isNode1_def: fixes Kl :: "('b + 'c) list set"   and lab :: "('b + 'c) list \<Rightarrow> ('d, 'b, 'c) F1 + 'e"   and kl :: "('b + 'c) list" shows "isNode1 Kl lab kl = (\<exists>x1. lab kl = Inl x1 \<and> F1set2 x1 = Succ1 Kl kl \<and> F1set3 x1 = Succ2 Kl kl)"
AE_I': fixes N :: "'b set"   and M :: "'b measure"   and P :: "'b \<Rightarrow> bool" assumes "N \<in> null_sets M"   and "{x \<in> space M. \<not> P x} \<subseteq> N" shows "almost_everywhere M P"
StrongEq_L_sym: fixes \<tau> :: "'b state \<times> 'b state"   and x :: "'b state \<times> 'b state \<Rightarrow> 'c"   and y :: "'b state \<times> 'b state \<Rightarrow> 'c" assumes "\<tau> \<Turnstile> x \<triangleq> y" shows "\<tau> \<Turnstile> y \<triangleq> x"
continuous_def: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'c" shows "continuous F f = (f \<longlongrightarrow> f (netlimit F)) F"
continuous_within_closed_nontrivial: fixes s :: "'a set"   and a :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "closed s"   and "a \<notin> s" shows "continuous (at a within s) f"
flowimg: shows "flow0 x ` (existence_ivl0 x \<inter> {0..}) = flow0 x ` {0..}"
cases: fixes a1 :: "nat"   and a2 :: "'b tensor" assumes "cprank_max a1 a2" obtains ds :: "nat list" where "a1 = 0" and "a2 = tensor0 ds"   | A :: "'b tensor" and B :: "'b tensor" and j :: "nat" where "a1 = Suc j" and "a2 = A + B" and "dims A = dims B" and "cprank_max1 A" and "cprank_max j B"
preferred_alts_altdef: fixes i :: "'agent"   and x :: "'alt" assumes "i \<in> agents" shows "preferred_alts (R i) x = {y \<in> alts. R i x y}"
dist_prod_def: fixes x :: "'e \<times> 'f"   and y :: "'e \<times> 'f" shows "dist x y = sqrt ((dist (fst x) (fst y))\<^sup>2 + (dist (snd x) (snd y))\<^sup>2)"