hcju/isabelleps huggingface dataset:
x_max: shows "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x"
line_integral_def: fixes F :: "'a \<Rightarrow> 'a"   and basis :: "'a set"   and g :: "real \<Rightarrow> 'a" shows "line_integral F basis g \<equiv> integral {0..1} (\<lambda>x. \<Sum>b\<in>basis. F (g x) \<bullet> b * (vector_derivative g (at x within {0..1}) \<bullet> b))"
abs_pat_eqvt: fixes pi :: "(name \<times> name) list"   and p :: "pat"   and t :: "btrm" shows "pi \<bullet> (\<lambda>[p]. t) = (\<lambda>[pi \<bullet> p]. pi \<bullet> t)"
hh: fixes e :: "nat" shows "eval (Cn 1 (Pr 1 Z g) [Cn 1 r_length [recf.Id 1 0], recf.Id 1 0]) [e] \<down>= (if \<forall>i<e_length e. e_nth e i < 2 then 0 else e_nth e (GREATEST i. i < e_length e \<and> 2 \<le> e_nth e i))"
prob_space_axioms: fixes M :: "'d pmf" shows "prob_space (measure_pmf M)"
cs_path_swap: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and \<pi>' :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "IFC prob"   and "IFC_def.is_path prob \<pi>"   and "IFC_def.is_path prob \<pi>'"   and "IFC_def.eq_up_to \<pi> k \<pi>'" shows "IFC_def.cs prob \<pi> k = IFC_def.cs prob \<pi>' k"
cases: fixes a :: "dB" assumes "NF a" obtains ts :: "dB list" and x :: "nat" where "a = Var x \<degree>\<degree> ts" and "listall NF ts"   | t :: "dB" where "a = Abs t" and "NF t"
evalRecvArgs_gState_progress_rel: fixes prog :: "program"   and g :: "gState\<^sub>I"   and rargs :: "recvArg list"   and xs :: "integer list"   and p :: "pState" assumes "gState_inv prog g" shows "(g, fst (evalRecvArgs rargs xs g p)) \<in> gState_progress_rel prog"
degree_notin_vars: fixes x :: "nat"   and B :: "'a mpoly" assumes "x \<notin> vars B" shows "MPoly_Type.degree B x = 0"
continuous_blinfun_componentwiseI1: fixes F :: "'e filter"   and f :: "'e \<Rightarrow> 'd \<Rightarrow>\<^sub>L 'f" assumes "\<And>i. i \<in> Basis \<Longrightarrow> continuous F (\<lambda>x. blinfun_apply (f x) i)" shows "continuous F f"
complementTA_correct_1: shows "ta_lang ta.complementTA = ranked_trees A - ta_lang TA"
nvalid_def: fixes n :: "nat"   and t :: "triple" shows "\<Turnstile>n: t \<equiv> let (P, c, Q) = t in \<forall>s t. s -c-n\<rightarrow> t \<longrightarrow> P s \<longrightarrow> Q t"
bounded_lattice_top_def: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and top :: "'c" shows "class.bounded_lattice_top inf less_eq less sup top \<equiv> class.lattice inf less_eq less sup \<and> class.order_top less_eq less top"
family_oracle_def: fixes f :: "'a \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'd) spmf"   and s :: "'a \<Rightarrow> 'd" shows "family_oracle f s = (\<lambda>(i, x). map_spmf (\<lambda>(y, s'). (y, s(i := s'))) (f i (s i) x))"
g_4: shows "inj_on g ({a..b} \<union> (s \<union> g -` s'))"
distinctness_def: fixes ES :: "('b \<times> 'c) set" shows "distinctness ES \<equiv> \<forall>X rhs rhs'. (X, rhs) \<in> ES \<and> (X, rhs') \<in> ES \<longrightarrow> rhs = rhs'"
set_iterator_map_rev_linord_def: fixes it :: "('d \<Rightarrow> bool) \<Rightarrow> ('a \<times> 'e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and S0 :: "('a \<times> 'e) set" shows "set_iterator_map_rev_linord it S0 \<equiv> set_iterator_genord it S0 (\<lambda>(k, uu_) (k', uu_). k' \<le> k)"
partition_onI: fixes P :: "'b set set"   and A :: "'b set" assumes "\<Union> P = A"   and "\<And>p q. \<lbrakk>p \<in> P; q \<in> P; p \<noteq> q\<rbrakk> \<Longrightarrow> disjnt p q"   and "{} \<notin> P" shows "partition_on A P"
q'_def: shows "q' = (\<Sum>i<m. monom (1::'a) (m - Suc i) * c (n + i))"
s1_s2: fixes q :: "nat"   and x :: "nat"   and a :: "nat"   and b :: "nat"   and r :: "nat"   and y :: "nat" assumes "secure_mult q"   and "x < q"   and "a < q"   and "b < q"   and "r < q"   and "y < q" shows "((x + a) mod q * b + q - (a * b + q - r) mod q) mod q = (x * y + q - (x * ((y + q - b) mod q) + q - r) mod q) mod q"
gcd_cong_r: fixes y :: "'a"   and y' :: "'a"   and x :: "'a" assumes "y \<sim> y'"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "y' \<in> carrier G" shows "somegcd G x y \<sim> somegcd G x y'"
satisfies_T0: shows "is_left_adjoint (tab\<^sub>0 (r \<star> s))"
uminus_carrier_iff_mat: fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat" shows "(- A \<in> carrier_mat nr nc) = (A \<in> carrier_mat nr nc)"
decode_minsky_state_def: fixes n :: "nat" shows "decode_minsky_state n = (set (decode_minsky (c_fst n)), decode_regs (c_snd n))"
S_list_not_min_gallery_deletion: fixes ss :: "'a permutation list" assumes "ss \<in> lists (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"   and "ss \<noteq> []"   and "\<not> min_gallery (map (\<lambda>w. w `\<rightarrow> C0) (sums ss))" shows "\<exists>a b as bs cs. ss = as @ [a] @ bs @ [b] @ cs \<and> sum_list ss = sum_list (as @ bs @ cs)"
out_exprChannelSingle_def: fixes sP :: "specID"   and ch :: "chanID"   and E :: "Expression" shows "out_exprChannelSingle sP ch E \<equiv> ch \<in> out sP \<and> exprChannel ch E \<and> (\<forall>x t. x \<in> out sP \<and> x \<noteq> ch \<longrightarrow> \<not> exprChannel x E)"
is_symbol_distinct: fixes x :: "symbol" assumes "is_symbol x" shows "is_terminal x \<noteq> is_nonterminal x"
one_add_one: fixes plus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and one :: "'f" assumes "class.numeral plus" shows "plus one one = numeral.numeral one plus (num.Bit0 num.One)"
product_is_ide: fixes J :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'b \<Rightarrow> 'a"   and a :: "'a" assumes "has_as_product J D a" shows "ide a"
linear_order: shows "linear_order {(x, y). x \<le> y}"
seqSubst2: fixes \<sigma> :: "(name list \<times> 'a list) list"   and x :: "name"   and P :: "('a, 'b, 'c) psi" assumes "wellFormedSubst \<sigma>"   and "x \<sharp> \<sigma>"   and "x \<sharp> P" shows "x \<sharp> P[<\<sigma>>]"
validity_def: fixes ES :: "('b list set \<times> 'b trm set) set" shows "validity ES \<equiv> \<forall>(X, rhs)\<in>ES. rhss rhs \<subseteq> lhss ES"
VInter_in_VsetI: fixes A :: "V"   and \<alpha> :: "V" assumes "A \<in>\<^sub>\<circ> Vset \<alpha>" shows "\<Inter>\<^sub>\<circ> A \<in>\<^sub>\<circ> Vset \<alpha>"
assms_1: shows "\<forall>(k, \<gamma>)\<in>boundary C. valid_path \<gamma>"
cfun_below_iff: fixes f :: "'d \<rightarrow> 'e"   and g :: "'d \<rightarrow> 'e" shows "(f \<sqsubseteq> g) = (\<forall>x. f\<cdot>x \<sqsubseteq> g\<cdot>x)"
rep_eq: fixes x :: "'a reg_lan" shows "Rep_reg_lan (x\<^sup>\<star>) = Rep_reg_lan x\<^sup>\<star>"
c: shows "(SUP n\<in>{1..}. Inf (u ` {n..})) = (SUP n. Inf (u ` {n..}))"
poly_no_roots_geq: fixes a :: "real"   and p :: "real poly" shows "(\<forall>x\<ge>a. poly p x \<noteq> 0) = (p \<noteq> 0 \<and> poly p a \<noteq> 0 \<and> count_roots_above p a = 0)"
element_image: fixes g :: "'a"   and x :: "'c"   and y :: "'c" assumes "g \<in> carrier G"   and "x \<in> E"   and "\<phi> g x = y" shows "y \<in> E"
simps_8: fixes p :: "pf"   and xs :: "nat list" shows "eval_pf (Neg p) xs = (\<not> eval_pf p xs)"
unord_resolve_sound: fixes C :: "'a literal multiset"   and D :: "'a literal multiset"   and E :: "'a literal multiset"   and I :: "'a set" assumes "unord_resolve C D E"   and "I \<Turnstile> C"   and "I \<Turnstile> D" shows "I \<Turnstile> E"
exact_aux_Nil_iff: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and k :: "nat"   and m :: "nat" assumes "valid_decomp X qs"   and "standard_decomp k qs"   and "exact_decomp m qs" shows "(exact_aux k m qs = []) = (qs = [])"
card_gcd_eq_totient: fixes n :: "nat"   and d :: "nat" assumes "0 < n"   and "d dvd n" shows "card {k \<in> {0<..n}. gcd k n = d} = totient (n div d)"
Res: fixes R :: "pi"   and S :: "pi"   and x :: "name" assumes "(R, S) \<in> Rel" shows "(<\<nu>x>R, <\<nu>x>S) \<in> Rel"
sheap_ok: shows "P,h \<turnstile>\<^sub>s sh \<surd>"
hyps: shows "cis (real n * 2 * pi) = 1"
frag_expansion: fixes a :: "'c \<Rightarrow>\<^sub>0 int" shows "a = frag_extend frag_of a"
leg1_composite: shows "H\<mu>H\<nu>\<pi>.leg1 = \<mu>.leg1 \<cdot> \<mu>_\<nu>\<pi>.prj\<^sub>1"
name_abs_eq: fixes a :: "name"   and x :: "'a"   and b :: "name"   and y :: "'a" shows "([a].x = [b].y) = (a = b \<and> x = y \<or> a \<noteq> b \<and> x = [(a, b)] \<bullet> y \<and> a \<sharp> y)"
diff_diff_add_mset: fixes M :: "'c multiset"   and N :: "'c multiset"   and P :: "'c multiset" shows "M - N - P = M - (N + P)"
np': shows "isnpolyh ((shift1 ^^ (Reflected_Multivariate_Polynomial.degree s - n)) p) 0"
std\<^sub>2_falsifies: fixes C\<^sub>2 :: "fterm literal set"   and G :: "bool list" shows "falsifies\<^sub>c G C\<^sub>2 = falsifies\<^sub>c G (C\<^sub>2 \<cdot>\<^sub>l\<^sub>s (\<lambda>x. \<epsilon> (''2'' @ x)))"
exec_move_While1: fixes pc :: "nat"   and e :: "'addr expr1"   and ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e' :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and xcp :: "'addr option" assumes "pc < length (compE2 e)" shows "exec_move ci P t (while (e) e') h (stk, loc, pc, xcp) = exec_move ci P t e h (stk, loc, pc, xcp)"
terminates_iff_terminates_normalize: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate" shows "\<Gamma>\<turnstile>Language.normalize c \<down> s = \<Gamma>\<turnstile>c \<down> s"
ncol_conga_ncol: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Col Bet A B C"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B C D E F" shows "\<not> Tarski_neutral_dimensionless.Col Bet D E F"
small_step_induct: fixes x1a :: "com"   and x1b :: "char list \<Rightarrow> val"   and x2a :: "com"   and x2b :: "char list \<Rightarrow> val"   and P :: "com \<Rightarrow> (char list \<Rightarrow> val) \<Rightarrow> com \<Rightarrow> (char list \<Rightarrow> val) \<Rightarrow> bool" assumes "(x1a, x1b) \<rightarrow> (x2a, x2b)"   and "\<And>a s v x. taval a s v \<Longrightarrow> P (x ::= a) s SKIP (s(x := v))"   and "\<And>c s. P (SKIP;; c) s c s"   and "\<And>c1 s c1' s' c2. \<lbrakk>(c1, s) \<rightarrow> (c1', s'); P c1 s c1' s'\<rbrakk> \<Longrightarrow> P (c1;; c2) s (c1';; c2) s'"   and "\<And>b s c1 c2. tbval b s True \<Longrightarrow> P (IF b THEN c1 ELSE c2) s c1 s"   and "\<And>b s c1 c2. tbval b s False \<Longrightarrow> P (IF b THEN c1 ELSE c2) s c2 s"   and "\<And>b c s. P (WHILE b DO c) s (IF b THEN c;; WHILE b DO c ELSE SKIP) s" shows "P x1a x1b x2a x2b"
Key_unique_SesKey: fixes K :: "nat"   and SesKey :: "nat"   and B :: "agent"   and T :: "msg"   and Ticket :: "msg"   and evs :: "event list"   and K' :: "nat"   and B' :: "agent"   and T' :: "msg"   and Ticket' :: "msg" assumes "Crypt K \<lbrace>Key SesKey, Agent B, T, Ticket\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt K' \<lbrace>Key SesKey, Agent B', T', Ticket'\<rbrace> \<in> parts (knows Spy evs)"   and "Key SesKey \<notin> analz (knows Spy evs)"   and "evs \<in> kerbIV" shows "K = K' \<and> B = B' \<and> T = T' \<and> Ticket = Ticket'"
a_subid': fixes x :: "'a" shows "ad x \<le> (1::'a)"
poly_mult_prepend_replicate_zero: fixes p1 :: "'a list"   and p2 :: "'a list"   and n :: "nat" assumes "set p1 \<subseteq> carrier R"   and "set p2 \<subseteq> carrier R" shows "poly_mult p1 p2 = poly_mult (replicate n \<zero> @ p1) p2"
zero_le_divide_abs_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and 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 sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq zero (divide a (abs b)) = (less_eq zero a \<or> b = zero)"
ltake_epred_ltl: fixes n :: "enat"   and xs :: "'c llist" shows "ltake (epred n) (ltl xs) = ltl (ltake n xs)"
simps_2: fixes \<phi>\<^sub>1 :: "atom fm"   and \<phi>\<^sub>2 :: "atom fm" shows "inf\<^sub>+ (Or \<phi>\<^sub>1 \<phi>\<^sub>2) = or (inf\<^sub>+ \<phi>\<^sub>1) (inf\<^sub>+ \<phi>\<^sub>2)"
S_def: shows "S \<equiv> \<lparr>g_V = UNIV, g_E = rel_of_enex (en, ex), g_V0 = {init}, sa_L = int\<rparr>"
IH: shows "fv\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>t \<delta>) \<subseteq> fv\<^sub>s\<^sub>t S \<union> range_vars \<delta>"
mem: shows "bl2 \<in># remove1_mset bl \<B>"
natural_isomorphism_axioms: shows "natural_isomorphism (\<cdot>) (\<cdot>) L local.map \<ll>.map"
isViewOn_def: fixes \<V> :: "'b V_rec"   and E :: "'b set" shows "isViewOn \<V> E \<equiv> V_valid \<V> \<and> V\<^bsub>\<V>\<^esub> \<union> N\<^bsub>\<V>\<^esub> \<union> C\<^bsub>\<V>\<^esub> = E"
cod_simps_3: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" assumes "identity_arrow_of_spans C \<mu>" shows "Leg1 (Cod \<mu>) = Leg1 (Dom \<mu>)"
poly_x_mult_y_hom: fixes p :: "'a poly poly" shows "poly_x_mult_y (map_poly (map_poly hom) p) = map_poly (map_poly (map_poly hom)) (poly_x_mult_y p)"
image_image: fixes f :: "'i \<Rightarrow> 'h"   and g :: "'j \<Rightarrow> 'i"   and A :: "'j set" shows "f ` g ` A = (\<lambda>x. f (g x)) ` A"
k_th_has_derivative_at_right: shows "((\<lambda>x. nth_derivative k f x 1) has_derivative (\<lambda>x. 0)) (at 0 within {0..})"
lunionM_lUnionM1: fixes A :: "'m list"   and x :: "'m" shows "lunionM (lUnionM A) x = foldr lunionM A x"
last_step: shows "c k \<Rightarrow>\<^sub>d c (Suc k)"
igSubstInpIPresIGWlsInpSTR_def: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) model_scheme" shows "igSubstInpIPresIGWlsInpSTR MOD \<equiv> \<forall>ys y Y delta inp. igWls MOD (asSort ys) Y \<longrightarrow> igWlsInp MOD delta (igSubstInp MOD ys Y y inp) = igWlsInp MOD delta inp"
s_UnauthI_inv: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and \<pi>' :: "Syntax.term list"   and e' :: "Syntax.term" assumes "\<lless>\<pi>, Unauth e\<ggreater> I\<rightarrow> \<lless>\<pi>', e'\<ggreater>" obtains e'a :: "Syntax.term" where "e' = Unauth e'a" and "\<lless>\<pi>, e\<ggreater> I\<rightarrow> \<lless>\<pi>', e'a\<ggreater>"   | "\<pi>' = \<pi>" and "e' = e" and "value e"
valid_union_conc: fixes fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and c :: "'a list multiset"   and c' :: "'a list multiset" shows "valid fg (c + c') = (valid fg c \<and> valid fg c' \<and> mon_c fg c \<inter> mon_c fg c' = {})"
eq: shows "emeasure N {x \<in> space M. RN_deriv M N x = \<infinity>} = \<infinity> * emeasure M {x \<in> space M. RN_deriv M N x = \<infinity>}"
rng_to_rng_of_frac_def: fixes R :: "('c, 'd) ring_scheme"   and S :: "'c set"   and r :: "'c" assumes "eq_obj_rng_of_frac R S" shows "eq_obj_rng_of_frac.rng_to_rng_of_frac R S r \<equiv> r |\<^bsub>\<lparr>carrier = carrier R \<times> S, eq = \<lambda>(r, s) (r', s'). \<exists>t\<in>S. t \<otimes>\<^bsub>R\<^esub> (s' \<otimes>\<^bsub>R\<^esub> r \<ominus>\<^bsub>R\<^esub> s \<otimes>\<^bsub>R\<^esub> r') = \<zero>\<^bsub>R\<^esub>\<rparr>\<^esub> \<one>\<^bsub>R\<^esub>"
finvimageE: fixes a :: "V"   and r :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> r -`\<^sub>\<bullet> A" obtains b :: "V" where "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r" and "b \<in>\<^sub>\<circ> A"
hom: fixes G :: "'a set"   and T :: "'a \<Rightarrow> 'a"   and g :: "'a"   and g' :: "'a" assumes "GroupEnd G T"   and "g \<in> G"   and "g' \<in> G" shows "T (g + g') = T g + T g'"
abs_ivl_def: shows "abs_ivl \<equiv> quot_type.abs eq_ivl Abs_ivl"
Remove_in_card_measure: fixes ES :: "('b list set \<times> 'b trm set) set"   and X :: "'b list set"   and rhs :: "'b trm set" assumes "finite ES"   and "(X, rhs) \<in> ES" shows "(Remove ES X rhs, ES) \<in> measure card"
assms: shows "low_equal s1 s2 \<and> t1 = snd (fst (add_instr instr s1)) \<and> t2 = snd (fst (add_instr instr s2))"
idem: fixes a :: "'b \<times> 'c" shows "inf_prod a a = a"
set_lebesgue_integral_def: fixes M :: "'d measure"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c" shows "set_lebesgue_integral M A f \<equiv> LINT x|M. indicat_real A x *\<^sub>R f x"
a1: shows "\<Gamma> \<turnstile> e\<^sub>1 : T'\<rightarrow>T"
assms_2: shows "z \<in> inside (path_image \<gamma>)"
rtrancl_list_impl: fixes r :: "('a \<times> 'a) list"   and as :: "'a list" shows "set (rtrancl_list_impl r as) = {b. \<exists>a\<in>set as. (a, b) \<in> (set r)\<^sup>*}"
norm_preserves_Can: fixes t :: "'c local.term" assumes "Can t" shows "Can \<^bold>\<parallel>t\<^bold>\<parallel>"
conf_clinit_diff': fixes P :: "jvm_prog"   and sh :: "sheap"   and stk :: "val list"   and loc :: "val list"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and stk' :: "val list"   and loc' :: "val list"   and pc' :: "nat" assumes "conf_clinit P sh ((stk, loc, C, M, pc, ics) # frs)" shows "conf_clinit P sh ((stk', loc', C, M, pc', No_ics) # frs)"
index_less_size_conv: fixes xs :: "'a list"   and x :: "'a" shows "(index xs x < length xs) = (x \<in> set xs)"
llist_all2_llist_of_stream: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c stream"   and ys :: "'d stream" shows "llist_all2 P (llist_of_stream xs) (llist_of_stream ys) = stream_all2 P xs ys"
OpCatCatExt: fixes C :: "('d, 'e, 'f) Category_scheme" assumes "ExtCategory C" shows "ExtCategory (Op C)"
max_correct_1: fixes s :: "'s"   and P :: "'u \<times> 'v \<Rightarrow> bool" assumes "invar s"   and "rel_of (\<alpha> s) P \<noteq> {}" shows "max s P \<in> Some ` rel_of (\<alpha> s) P"
flrestriction_vempty: fixes A :: "V" shows "[]\<^sub>\<circ> \<restriction>\<^sup>l\<^sub>\<bullet> A = []\<^sub>\<circ>"
finite_psubset_induct: fixes A :: "'a set"   and P :: "'a set \<Rightarrow> bool" assumes "finite A"   and "\<And>A. \<lbrakk>finite A; \<And>B. B \<subset> A \<Longrightarrow> P B\<rbrakk> \<Longrightarrow> P A" shows "P A"
edom_join: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" shows "edom (f \<squnion> g) = edom f \<union> edom g"
81: fixes x :: "'a" shows "top \<squnion> - x = top"
antisym: fixes x :: "'a set"   and y :: "'a set" assumes "x \<sqsubseteq>\<^sup>s y"   and "y \<sqsubseteq>\<^sup>s x"   and "x \<in> Q"   and "y \<in> Q" shows "x = y"
broadcasts_unique: fixes history :: "nat \<Rightarrow> 'c event list"   and msg_id :: "'c \<Rightarrow> 'd"   and i :: "nat"   and j :: "nat"   and m :: "'c" assumes "network history msg_id"   and "i \<noteq> j"   and "Broadcast m \<in> set (history i)" shows "Broadcast m \<notin> set (history j)"
\<eta>': shows "\<a>[g, f\<^sub>a, f] \<cdot> (\<nu> \<star> f) \<cdot> \<rho> \<cdot> \<r>[g] = g \<star> \<eta> \<and> (\<l>[f] \<cdot> (\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, f\<^sub>a, f]) \<cdot> (f \<star> \<eta>) = \<r>[f]"
scaleR_mono': 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 scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "real"   and b :: "real"   and c :: "'a"   and d :: "'a" assumes "class.ordered_real_vector plus zero minus uminus less_eq less scaleR"   and "a \<le> b"   and "less_eq c d"   and "0 \<le> a"   and "less_eq zero c" shows "less_eq (scaleR a c) (scaleR b d)"
gp_fin: shows "AE x in M. ennreal (f x) * nn_cond_exp M F gp x \<noteq> \<infinity>"
convex_real_interval_3: fixes a :: "real" shows "convex {a<..}"
agree_on_refl: fixes R :: "('b, 'c) labeled_graph"   and f :: "('c \<times> 'd) set" shows "agree_on R f f"
subring_is_ring: fixes H :: "'a set" assumes "subring H (L\<lparr>carrier := A\<rparr>)" shows "ring (L\<lparr>carrier := A, carrier := H\<rparr>)"
Pair_Collect: fixes x :: "'a"   and y :: "'b"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "((x, y) \<in> {(x, y). P x y}) = P x y"
Uniq_def: fixes P :: "'a \<Rightarrow> bool" shows "Uniq P \<equiv> \<forall>x y. P x \<longrightarrow> P y \<longrightarrow> y = x"
pib: shows "p * A $ a $ j + q * A $ b $ j = d \<and> d dvd A $ a $ j \<and> d dvd A $ b $ j \<and> (\<forall>d'. d' dvd A $ a $ j \<and> d' dvd A $ b $ j \<longrightarrow> d' dvd d) \<and> d * u = - A $ b $ j \<and> d * v = A $ a $ j"
disjoint_Nv2_Vv1: shows "N\<^bsub>\<V>2\<^esub> \<inter> V\<^bsub>\<V>1\<^esub> = {}"
small_set_vdiff: fixes A :: "V"   and B :: "V" shows "small {x. x \<in>\<^sub>\<circ> A \<and> x \<notin>\<^sub>\<circ> B}"
convex_closed_interval: fixes a :: "'b"   and b :: "'b" shows "convex {a..b}"
assms: shows "x \<in> vars\<^sub>t (t \<cdot>\<^sub>t (\<lambda>x. \<epsilon> (y @ x)))"
subgame_E: fixes G :: "('c, 'd) ParityGame_scheme"   and V' :: "'c set" assumes "ParityGame G" shows "E\<^bsub>ParityGame.subgame G V'\<^esub> \<subseteq> E\<^bsub>G\<^esub>"
arc_rectangle: fixes x :: "'a" assumes "bijective (x * top) \<and> bijective (x[[top]])" shows "rectangle x"
lossless_res: fixes \<eta> :: "nat" assumes "lossless" shows "lossless_resource (\<I>_res \<eta> \<oplus>\<^sub>\<I> \<I>_common' \<eta>) (res \<eta>)"
simps_1: fixes l :: "'c tree"   and v :: "'c"   and r :: "'c tree" shows "right \<langle>l, v, r\<rangle> = r"
axioms_3: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> nat"   and parOf :: "'b \<Rightarrow> nat"   and \<Phi> :: "('a, 'b) lit list set"   and D :: "'c \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> 'c list \<Rightarrow> 'c"   and intP :: "'b \<Rightarrow> 'c list \<Rightarrow> bool" assumes "U.Model wtFsym wtPsym arOf parOf \<Phi> D intF intP" shows "U.Model_axioms \<Phi> D intF intP"
Tinv_def: shows "I.Tinv \<equiv> inv Tinv"
simps_18: shows "cospan_shape.comp cospan_shape.AT cospan_shape.BB = cospan_shape.Null"
assms: shows "nsqn\<^sub>r r < nsqn\<^sub>r r' \<or> nsqn\<^sub>r r = nsqn\<^sub>r r' \<and> \<pi>\<^sub>5 r' \<le> \<pi>\<^sub>5 r"
inverse_arrows_def: fixes f :: "'a2"   and g :: "'a2" shows "C2.inverse_arrows f g \<equiv> C2.ide (g \<cdot>\<^sub>2 f) \<and> C2.ide (f \<cdot>\<^sub>2 g)"
induced_arrowI_1: fixes \<chi>' :: "(bool, unit) J.arr \<Rightarrow> 'c"   and a' :: "'c" assumes "\<chi>' \<in> D.cones a'" shows "\<guillemotleft>induced_arrow a' \<chi>' : a' \<rightarrow> C.dom p0\<guillemotright>"
zero_less_divide_iff: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sgn :: "'e \<Rightarrow> 'e"   and abs :: "'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less zero (divide a b) = (less zero a \<and> less zero b \<or> less a zero \<and> less b zero)"
eint_mult_mono: fixes c :: "eint"   and n :: "eint"   and k :: "eint" assumes "0 < c \<and> c \<noteq> \<infinity>"   and "n < k" shows "n * c < k * c"
MakeFtor: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme" assumes "FunctorM F" shows "Functor (MakeFtor F)"
simps_2: fixes l :: "('b \<times> nat) tree"   and uu :: "'b"   and n :: "nat"   and r :: "('b \<times> nat) tree" shows "wbt \<langle>l, (uu, n), r\<rangle> = (n = size l + size r + 1 \<and> balanced1 l r \<and> balanced1 r l \<and> wbt l \<and> wbt r)"