hcju/isabelleps huggingface dataset:
inline1_inline_conv_inline2': fixes callee1 :: "'f \<Rightarrow> 'k \<Rightarrow> ('l \<times> 'f, 'g, 'h) gpv"   and callee2 :: "'e \<Rightarrow> 'j \<Rightarrow> ('i \<times> 'e, 'k, 'l) gpv"   and gpv :: "('d, 'j, 'i) gpv"   and s2 :: "'e"   and s1 :: "'f" shows "inline1 (\<lambda>(s2, s1) c2. map_gpv (\<lambda>((r, s2), s1). (r, s2, s1)) id (inline callee1 (callee2 s2 c2) s1)) gpv (s2, s1) = map_spmf (map_sum id (\<lambda>(x, rpv'', rpv', rpv). (x, \<lambda>r. rpv'' r \<bind> (\<lambda>(r1, s1). map_gpv (\<lambda>((r2, s2), s1). (r2, s2, s1)) id (inline callee1 (rpv' r1) s1)), rpv))) (inline2 callee1 callee2 gpv s2 s1)"
coplanar_small: fixes s :: "'a set" assumes "finite s"   and "card s \<le> 3" shows "coplanar s"
Tinv_mpt: assumes "invertible_qmpt" shows "mpt M Tinv"
is_UP_monom_def: shows "is_UP_monom = (\<lambda>f. f \<in> carrier (UP R) \<and> f = monom P (f (deg R f)) (deg R f))"
less_option_None: fixes x :: "'a option" shows "(x < None) = False"
card_greater: fixes A :: "'c set"   and Q :: "'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool" assumes "finite A"   and "card {x \<in> A. Q x} < card {x \<in> A. P x}" obtains C :: "'c set" where "card ({x \<in> A. P x} - C) = card {x \<in> A. Q x}" and "C \<noteq> {}" and "C \<subseteq> {x \<in> A. P x}"
opt_bind_cong: fixes f :: "'b \<Rightarrow> 'c option"   and f' :: "'b \<Rightarrow> 'c option"   and g :: "'c \<Rightarrow> 'b \<Rightarrow> 'd option"   and g' :: "'c \<Rightarrow> 'b \<Rightarrow> 'd option" assumes "f = f'"   and "\<And>v s. f' s = Some v \<Longrightarrow> g v s = g' v s" shows "f |>> g = f' |>> g'"
raw_converter_invariant: fixes \<I>' :: "('g, 'h) \<I>" shows "raw_converter_invariant \<I> \<I>' (\<lambda>s x. lift_spmf (callee s x)) I"
ex_un_null: shows "\<exists>!n. \<forall>f. n \<cdot>\<^sub>S f = n \<and> f \<cdot>\<^sub>S n = n"
minus_mult_div_eq_mod: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_modulo plus minus zero times one divide modulo" shows "minus a (times b (divide a b)) = modulo a b"
abc_bcd_acd: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "MinkowskiBetweenness \<E> \<P> betw"   and "betw a b c"   and "betw b c d" shows "betw a c d"
tcontra_halt: assumes "halts tcontra [code_tcontra]" shows "False"
indep_vars_iff_distr_eq_PiM': fixes M :: "'a measure"   and I :: "'b set"   and X :: "'b \<Rightarrow> 'a \<Rightarrow> 'c"   and M' :: "'b \<Rightarrow> 'c measure" assumes "prob_space M"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> X i \<in> M \<rightarrow>\<^sub>M M' i" shows "prob_space.indep_vars M M' X I = (distr M (Pi\<^sub>M I M') (\<lambda>x. \<lambda>i\<in>I. X i x) = Pi\<^sub>M I (\<lambda>i. distr M (M' i) (X i)))"
loop_stepA: fixes P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and V :: "'a set"   and u :: "'a" assumes "inv\<^sub>3 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V"   and "B\<^sub>1 = {}"   and "V \<inter> L = {}"   and "u \<in> V \<inter> S" shows "inv\<^sub>3 P\<^sub>1 P\<^sub>2 {u} B\<^sub>2 (V - {u})"
gfinprod_cong: fixes A :: "'f set"   and B :: "'f set"   and f :: "'f \<Rightarrow> 'a"   and g :: "'f \<Rightarrow> 'a" assumes "A = B"   and "f \<in> B \<rightarrow> carrier G"   and "\<And>i. i \<in> B =simp=> f i = g i" shows "gfinprod G f A = gfinprod G g B"
card_of_card_order_on: fixes A :: "'b set" shows "card_order_on A |A|"
a_closed: fixes x :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int"   and y :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int" assumes "x \<in> carrier (function_ring (carrier Zp) Zp)"   and "y \<in> carrier (function_ring (carrier Zp) Zp)" shows "x \<oplus>\<^bsub>function_ring (carrier Zp) Zp\<^esub> y \<in> carrier (function_ring (carrier Zp) Zp)"
boundedTreeInduction: fixes subs :: "'a \<Rightarrow> 'a set"   and Gamma :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "fans subs"   and "bounded (tree subs Gamma)"   and "founded subs P (tree subs Gamma)"   and "\<forall>delta. \<not> terminal subs delta \<longrightarrow> (\<forall>sigma\<in>subs delta. P sigma) \<longrightarrow> P delta" shows "P Gamma"
nonBVG_rule: fixes \<alpha> :: "game"   and x :: "variable" assumes "\<And>I \<omega> X. (\<omega> \<in> game_sem I \<alpha> X) = (\<omega> \<in> game_sem I \<alpha> (selectlike X \<omega> {x}))" shows "x \<notin> BVG \<alpha>"
1: fixes x :: "'a prefix_match option"   and y :: "'a wordinterval"   and x2 :: "'a prefix_match" assumes "\<not> wordinterval_empty rs"   and "(x, y) = wordinterval_CIDR_split1 rs"   and "x = Some x2" shows "\<forall>e\<in>set (wordinterval_CIDR_split_prefixmatch y). valid_prefix e \<and> pfxm_length e \<le> LENGTH('a)"
elims_3: fixes x :: "'b mat" assumes "\<not> square_mat x" obtains A :: "'b mat" where "x = A" and "dim_col A \<noteq> dim_row A"
ann_SEM_def: fixes c :: "'b ann_com"   and S :: "'b set" shows "ann_SEM c S \<equiv> \<Union> (ann_sem c ` S)"
evals_cases_2: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and es :: "char list exp list"   and s :: "heap \<times> sfields \<times> sheap"   and e' :: "char list exp list"   and s' :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>e # es,s\<rangle> [\<Rightarrow>] \<langle>e',s'\<rangle>" obtains v :: "val" and s\<^sub>1 :: "heap \<times> sfields \<times> sheap" and es' :: "char list exp list" where "e' = Val v # es'" and "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1\<rangle>" and "P \<turnstile> \<langle>es,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>es',s'\<rangle>"   | e'a :: "char list exp" where "e' = throw e'a # es" and "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>throw e'a,s'\<rangle>"
osp__ncop1: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and P :: "'p"   and Q :: "'p" assumes "A B C OSP P Q" shows "\<not> Coplanar A B C P"
Derives1_drop: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and a :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and b :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "CFG.Derives1 \<NN> \<TT> \<RR> a i r b" shows "drop (Suc i) a = drop (i + length (snd r)) b"
simps_2: fixes l :: "interval_tree"   and it :: "nat idx_interval"   and m :: "nat"   and r :: "interval_tree" shows "tree_sorted (Node l it m r) = ((\<forall>x\<in>tree_set l. x < it) \<and> Ball (tree_set r) ((<) it) \<and> tree_sorted l \<and> tree_sorted r)"
deadlock_free_v2_FD: fixes P :: "'a process" shows "deadlock_free_v2 P = DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P UNIV \<sqsubseteq>\<^sub>F\<^sub>D P"
simps_1: shows "gauss_jordan_main A B i j = (let nr = dim_row A; nc = dim_col A in if i < nr \<and> j < nc then let aij = A $$ (i, j) in if aij = (0::'a) then case concat (map (\<lambda>i'. if A $$ (i', j) \<noteq> (0::'a) then [i'] else []) [Suc i..<nr]) of [] \<Rightarrow> gauss_jordan_main A B i (Suc j) | i' # x \<Rightarrow> gauss_jordan_main (swaprows i i' A) (swaprows i i' B) i j else if aij = (1::'a) then let v = \<lambda>i. A $$ (i, j) in gauss_jordan_main (eliminate_entries v A i j) (eliminate_entries v B i j) (Suc i) (Suc j) else let iaij = inverse aij in gauss_jordan_main (multrow i iaij A) (multrow i iaij B) i j else (A, B))"
construct_NofI_prop: fixes p :: "real poly"   and I :: "real poly list" assumes "p \<noteq> 0" shows "construct_NofI p I = rat_of_int (int (card {x. poly p x = 0 \<and> 0 < poly (prod_list I) x}) - int (card {x. poly p x = 0 \<and> poly (prod_list I) x < 0}))"
zero_imp_mult_zero: fixes a :: "'a"   and b :: "'a" assumes "a = (0::'a) \<or> b = (0::'a)" shows "a * b = (0::'a)"
normalize_eq_0_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "(normalize a = zero) = (a = zero)"
vconn_implies_hpar_1: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.in_hom \<mu> f g" shows "Span.src \<mu> = Span.src f"
axioms_5: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b"   and a :: "'b"   and \<chi> :: "'a \<Rightarrow> 'b" assumes "Limit.cone J C D a \<chi>" shows "natural_transformation J C (constant_functor.map J C a) D \<chi>"
denote_while_n_iter_assoc: fixes M0 :: "complex mat"   and M1 :: "complex mat"   and DS :: "complex mat \<Rightarrow> complex mat"   and n :: "nat"   and \<rho> :: "complex mat" shows "denote_while_n_iter M0 M1 DS (Suc n) \<rho> = DS (M1 * denote_while_n_iter M0 M1 DS n \<rho> * adjoint M1)"
constr: fixes ts :: "value list"   and us :: "value list"   and name :: "name" assumes "list_all2 vrelated ts us" shows "\<turnstile>\<^sub>v Vconstr name ts \<approx> Vconstr name us"
eq_onp_le_eq: fixes P :: "'c \<Rightarrow> bool" shows "eq_onp P \<le> (=)"
cases: fixes a1 :: "'f list multiset"   and a2 :: "('g, 'h) label"   and a3 :: "'f list multiset"   and fg :: "('f, 'g, 'h, 'i, 'j) flowgraph_rec_scheme" assumes "(a1, a2, a3) \<in> refpoint fg" obtains u :: "'f" and a :: "'h" and v :: "'f" and r :: "'f list" and c :: "'f list multiset" where "a1 = add_mset (u__ # r__) cb__" and "a2 = LBase a" and "a3 = add_mset (v__ # r__) cb__" and "(u__, Base a, v__) \<in> edges fg" and "valid fg ({#u__ # r__#} + cb__)"   | u :: "'f" and p :: "'g" and v :: "'f" and r :: "'f list" and c :: "'f list multiset" where "a1 = add_mset (ua__ # ra__) cc__" and "a2 = LCall p__" and "a3 = add_mset (entry fg p__ # va__ # ra__) cc__" and "(ua__, Call p__, va__) \<in> edges fg" and "valid fg ({#ua__ # ra__#} + cc__)" and "mon fg p__ \<inter> mon_c fg cc__ = {}"   | p :: "'g" and r :: "'f list" and c :: "'f list multiset" where "a1 = add_mset (return fg pa__ # rb__) cd__" and "a2 = LRet" and "a3 = add_mset rb__ cd__" and "valid fg ({#return fg pa__ # rb__#} + cd__)"   | u :: "'f" and p :: "'g" and v :: "'f" and r :: "'f list" and c :: "'f list multiset" where "a1 = add_mset (u # r) c" and "a2 = LSpawn p" and "a3 = add_mset (v # r) (add_mset [entry fg p] c)" and "(u, Spawn p, v) \<in> edges fg" and "valid fg (add_mset (u # r) c)"
card_1_singletonE: fixes A :: "'a set" assumes "card A = 1" obtains x :: "'a" where "A = {x}"
f''_def: shows "f'' \<equiv> of_int_poly (Mp f)"
aforms_err_to_aforms_Nil: fixes n :: "nat" shows "aforms_err_to_aforms n [] = []"
ground_resolution_with_selection_axioms: fixes M :: "'a literal multiset set" shows "ground_resolution_with_selection (S_M S M)"
assms_1: fixes a :: "'c" assumes "a \<in> S" shows "f a \<in> carrier R"
nonzero_pow_emp: fixes m :: "nat"   and u :: "'b list" assumes "m \<noteq> 0" shows "(u \<^sup>@ m = \<epsilon>) = (u = \<epsilon>)"
distinct_7: shows "AsmNoWrite \<noteq> GuarNoReadOrWrite"
add_commute: fixes a :: "('a \<times> 'a) set"   and b :: "('a \<times> 'a) set" assumes "valid_frac a"   and "valid_frac b" shows "add_rel a b = add_rel b a"
exhaust: fixes y :: "eint" obtains int :: "int" where "y = eint int"   | "y = \<infinity>"
remove_cycles_cycles: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list" assumes "x \<in> set xs" shows "\<exists>xxs as. as @ concat (map ((#) x) xxs) @ remove_cycles xs x ys = xs \<and> x \<notin> set as"
strict_implies_not_eq: fixes a :: "'d DBMEntry"   and b :: "'d DBMEntry" assumes "a \<prec> b" shows "a \<noteq> b"
eq_Max_iff: fixes A :: "('x \<Rightarrow>\<^sub>0 nat) set"   and m :: "'x \<Rightarrow>\<^sub>0 nat" assumes "finite A"   and "A \<noteq> {}" shows "(m = ordered_powerprod_lin.Max A) = (m \<in> A \<and> (\<forall>a\<in>A. a \<preceq> m))"
is_normal_Float_def: fixes x :: "('a, 'b) IEEE.float itself"   and f :: "Float.float" shows "is_normal_Float x f = (mantissa f \<noteq> 0 \<and> bitlen \<bar>mantissa f\<bar> \<le> int (LENGTH('b) + 1) \<and> - int (bias x) - bitlen \<bar>mantissa f\<bar> + 1 < Float.exponent f \<and> Float.exponent f < 2 ^ LENGTH('a) - bitlen \<bar>mantissa f\<bar> - int (bias x))"
remove_vertex_Digraph: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c" assumes "Digraph G" shows "Digraph (Digraph.remove_vertex G v)"
11: fixes k :: "nat" assumes "\<tau> \<sigma>' k \<le> \<tau> \<sigma>' i + b" shows "k < Monitor.progress \<sigma> \<phi>1 (plen \<pi>)"
unique_rootE: fixes plr :: "int poly \<times> rat \<times> rat" assumes "\<exists>!x. root_cond plr x" obtains "real_of_rat (rai_lb plr) \<le> real_of_1 plr" and "real_of_1 plr \<le> real_of_rat (rai_ub plr)" and "poly (real_of_int_poly (poly_real_alg_1 plr)) (real_of_1 plr) = 0" and "root_cond plr (real_of_1 plr)" and "\<And>y. (real_of_1 plr = y) = root_cond plr y" and "\<And>y. (y = real_of_1 plr) = root_cond plr y"
turn_true_def: fixes \<A> :: "'b \<Rightarrow> bool"   and F :: "'b formula" shows "turn_true \<A> F \<equiv> if \<A> \<Turnstile> F then F else \<^bold>\<not> F"
Units: shows "carrier (DirProds G I) \<subseteq> Units (DirProds G I)"
x0_int: shows "real (nat \<lfloor>x\<^sub>0\<rfloor>) = x\<^sub>0"
distinc: fixes H :: "'a set"   and H1 :: "'a set"   and K :: "'a set"   and K1 :: "'a set" assumes "subgroup H G"   and "H1 \<lhd> G\<lparr>carrier := H\<rparr>"   and "subgroup K G"   and "K1 \<lhd> G\<lparr>carrier := K\<rparr>" shows "subgroup (H \<inter> K) (G\<lparr>carrier := normalizer G (H1 <#> H \<inter> K1)\<rparr>)"
a_prop: shows "[\<lparr>A!,a\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall>F. \<lbrace>a\<^sup>P,F\<rbrace> \<^bold>\<equiv> F \<^bold>= F) in v]"
Units_closed: fixes x :: "'a" assumes "x \<in> Units (mult_of R)" shows "x \<in> carrier (mult_of R)"
boundedI: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b"   and "a \<le> c" shows "a \<le> min b c"
monoid_comm_monoidI: assumes "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes> y = y \<otimes> x" shows "Group.comm_monoid G"
cod_trg_eq_dom_trg: shows "\<eta>.cod.trg = \<eta>.dtrg"
convex_on_ereal_sum_aux: fixes a :: "real"   and c :: "real"   and b :: "ereal" assumes "0 < 1 - a" shows "(1 - ereal a) * (ereal (c / (1 - a)) * b) = ereal c * b"
Landau_inequality: fixes f :: "complex poly" shows "mahler_measure_poly f \<le> l2norm_complex f"
extended_predist_le1: fixes x :: "'a Gromov_completion"   and y :: "'a Gromov_completion" shows "extended_predist x y \<le> 1"
prems: shows "wbt \<langle>l, (a, n), r\<rangle>"
assms: shows "\<forall>x. var_asm_not_written mds x \<longrightarrow> mem' x = mem x \<and> dma mem' x = dma mem x"
arith_extra_simps_19: fixes a :: "'a" shows "a * (0::'a) = (0::'a)"
m1_inv2i_servI: fixes x :: "'a m1r_state_scheme" assumes "\<And>A B Ra Na Kab nla. \<lbrakk>A \<notin> bad; runs x Ra = Some (Init, [A, B], aKey Kab # nla); Na = Ra $ na\<rbrakk> \<Longrightarrow> \<exists>Rs. Kab = sesK (Rs $ sk) \<and> runs x Rs = Some (Serv, [A, B], [aNon Na])" shows "x \<in> m1_inv2i_serv"
no_elt: shows "\<not> (\<exists>w\<in>X. [[w a b]] \<or> [[b c w]])"
length_sequences: fixes key :: "'b \<Rightarrow> 'c"   and xs :: "'b list" shows "length (sequences key xs) \<le> length xs"
stakeWhile_def: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c stream" shows "stakeWhile P xs = ltakeWhile P (llist_of_stream xs)"
measurable_linorder_from_keys_restrict': fixes A :: "'b set"   and B :: "'b set" assumes "finite A"   and "A \<subseteq> B" shows "linorder_from_keys A \<in> Pi\<^sub>M A (\<lambda>_. borel) \<rightarrow>\<^sub>M count_space (Pow (B \<times> B))"
right_unique_Grp: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "right_unique (BNF_Def.Grp A f)"
distinct_1: fixes x2 :: "'d" shows "None \<noteq> Some x2"
subst_cls_eq_grounding_of_cls_subset_eq: fixes D :: "'a literal multiset"   and \<sigma> :: "'s"   and C :: "'a literal multiset" assumes "D \<cdot> \<sigma> = C" shows "grounding_of_cls C \<subseteq> grounding_of_cls D"
even_prime_nat: fixes p :: "nat" assumes "prime p"   and "even p" shows "p = 2"
permutes_natset_ge: fixes p :: "'a \<Rightarrow> 'a"   and S :: "'a set" assumes "p permutes S"   and "\<forall>i\<in>S. i \<le> p i" shows "p = id"
sum_le_included: fixes s :: "'c set"   and t :: "'d set"   and g :: "'d \<Rightarrow> 'e multiset"   and i :: "'d \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> 'e multiset" assumes "finite s"   and "finite t"   and "\<forall>y\<in>t. {#} \<subseteq># g y"   and "\<forall>x\<in>s. \<exists>y\<in>t. i y = x \<and> f x \<subseteq># g y" shows "comm_monoid_add.sum (+) {#} f s \<subseteq># comm_monoid_add.sum (+) {#} g t"
inv_4: shows "local.inv_from_bot (\<lambda>A i. local.one_zero A i j) A (Suc i)"
fun_lub_def: fixes L :: "'c set \<Rightarrow> 'b"   and A :: "('a \<Rightarrow> 'c) set" shows "fun_lub L A = (\<lambda>x. L {y. \<exists>f\<in>A. y = f x})"
max_eqR: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b" shows "max a b = b"
scale_eq_0_iff: fixes a :: "'c"   and x :: "'c" shows "(a * x = (0::'c)) = (a = (0::'c) \<or> x = (0::'c))"
matchC_ZO_def: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and theta :: "(('b, 'c) com \<times> ('b, 'c) com) set"   and c :: "('b, 'c) com"   and d :: "('b, 'c) com" assumes "PL_Indis indis" shows "PL_Indis.matchC_ZO tval aval indis theta c d \<equiv> \<forall>s t c' s'. (s, t) \<in> indis \<and> PL.transC tval aval (c, s) (c', s') \<longrightarrow> (s', t) \<in> indis \<and> (c', d) \<in> theta \<or> (\<exists>d' t'. PL.transC tval aval (d, t) (d', t') \<and> (s', t') \<in> indis \<and> (c', d') \<in> theta) \<or> (\<exists>t'. PL.transT tval aval (d, t) t' \<and> (s', t') \<in> indis \<and> PL_Indis.discr tval aval indis c')"
fixespointwise_im: fixes f :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and B :: "'b set" assumes "fixespointwise f A"   and "order.greater_eq A B" shows "f ` B = B"
assms: shows "xs\<^sub>1 @ xs\<^sub>2 = ys\<^sub>1 @ ys\<^sub>2"
simps: fixes a :: "'a list" shows "sortedP a = (a = [] \<or> (\<exists>ys x. a = x # ys \<and> (\<forall>y\<in>set ys. le x y) \<and> sortedP ys))"
simps_5: fixes V :: "'a set"   and s :: "('b, 'a) Term.term"   and t :: "('b, 'a) Term.term"   and S :: "('b, 'a) strand_step list" shows "wf\<^sub>s\<^sub>t V (\<langle>check: s \<doteq> t\<rangle>\<^sub>s\<^sub>t # S) = wf\<^sub>s\<^sub>t V S"
le_add_diff_inverse: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and b :: "'c"   and a :: "'c" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq b a" shows "plus b (minus a b) = a"
consistentAppendElement: fixes v :: "Literal list"   and l :: "Literal" assumes "consistent v"   and "\<not> literalFalse l v" shows "consistent (v @ [l])"
tendsto_sandwich_mono: fixes f :: "real \<Rightarrow> real"   and c :: "real" assumes "(\<lambda>n. f (real n)) \<longlonglongrightarrow> c"   and "\<forall>\<^sub>F x in at_top. \<forall>y z. x \<le> y \<and> y \<le> z \<longrightarrow> f y \<le> f z" shows "(f \<longlongrightarrow> c) at_top"
lm108: fixes Z :: "'a set"   and X :: "'a set"   and Y :: "'a set" assumes "Z \<subseteq> X \<union> Y" shows "sum (X <| Y) (Z - X) = 0"
assignT: fixes \<xi> :: "'e"   and l :: "'h"   and u :: "'e \<Rightarrow> 'e"   and p :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" shows "((\<xi>, {l}\<lbrakk>u\<rbrakk> p), \<tau>, u \<xi>, p) \<in> seqp_sos \<Gamma>"
list_of_oalist_tl: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and list_of_oalist :: "'c \<Rightarrow> ('b \<times> 'd) list \<times> 'a"   and oalist_of_list :: "('b \<times> 'd) list \<times> 'a \<Rightarrow> 'c"   and xs :: "'c" assumes "oalist_abstract rep_key_order list_of_oalist oalist_of_list" shows "list_of_oalist (oalist_abstract.tl list_of_oalist oalist_of_list xs) = oalist_raw.tl_raw (list_of_oalist xs)"
len_array: fixes ar :: "'b tree \<times> nat" assumes "case ar of (t, l) \<Rightarrow> braun t \<and> l = size t" shows "(case ar of (t, l) \<Rightarrow> l) = length (case ar of (t, l) \<Rightarrow> list t)"
res_def: fixes range :: "'c set"   and mac_alg :: "'c \<Rightarrow> 'b \<Rightarrow> 'a spmf" shows "macode.res range mac_alg = 1\<^sub>C |\<^sub>= macode.enm mac_alg |\<^sub>= macode.dem mac_alg \<rhd> 1\<^sub>C |\<^sub>= parallel_wiring \<rhd> parallel_resource1_wiring \<rhd> rorc.res range \<parallel> MAC.INSEC.res"
geodesic_segment_between_def: fixes G :: "'b set"   and x :: "'b"   and y :: "'b" shows "geodesic_segment_between G x y = (\<exists>g. g 0 = x \<and> g (dist x y) = y \<and> isometry_on {0..dist x y} g \<and> G = g ` {0..dist x y})"
nres_correctD_2: fixes S :: "'a dres"   and \<Phi> :: "'a \<Rightarrow> bool" assumes "nres_of S \<le> SPEC \<Phi>" shows "S \<noteq> dFAIL"
deduct_1: fixes ps :: "('a, 'b) fm list"   and p :: "('a, 'b) fm"   and q :: "('a, 'b) fm" assumes "ps \<turnstile> p \<^bold>\<longrightarrow> q" shows "p # ps \<turnstile> q"
map_cong0: fixes x :: "('d, 'e, 'f) F2"   and f1 :: "'d \<Rightarrow> 'g"   and g1 :: "'d \<Rightarrow> 'g"   and f2 :: "'e \<Rightarrow> 'h"   and g2 :: "'e \<Rightarrow> 'h"   and f3 :: "'f \<Rightarrow> 'i"   and g3 :: "'f \<Rightarrow> 'i" assumes "\<And>z1. z1 \<in> F2set1 x \<Longrightarrow> f1 z1 = g1 z1"   and "\<And>z2. z2 \<in> F2set2 x \<Longrightarrow> f2 z2 = g2 z2"   and "\<And>z3. z3 \<in> F2set3 x \<Longrightarrow> f3 z3 = g3 z3" shows "F2map f1 f2 f3 x = F2map g1 g2 g3 x"
stopType_empty: fixes a :: "nat \<Rightarrow> stopType list"   and t :: "nat" assumes "msg (Suc 0) a"   and "a t \<noteq> [stop_vc]" shows "a t = []"
eSubst_simp4: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and ys :: "'varSort"   and eY :: "'c withERR"   and y :: "'var" shows "eSubst MOD ys eY y ERR = ERR"
split: fixes P :: "'f \<Rightarrow> bool"   and f1 :: "'f"   and f2 :: "'g \<Rightarrow> 'g list \<Rightarrow> 'f"   and list :: "'g list" shows "P (case list of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = ((list = [] \<longrightarrow> P f1) \<and> (\<forall>x21 x22. list = x21 # x22 \<longrightarrow> P (f2 x21 x22)))"
wf_infinite_down_chain_compatible: fixes R :: "('a \<times> 'a) set"   and f :: "nat \<Rightarrow> 'a"   and S :: "('a \<times> 'a) set" assumes "wf R"   and "\<forall>i. (f (Suc i), f i) \<in> R \<union> S"   and "R O S \<subseteq> R" shows "\<exists>k. \<forall>i. (f (Suc (i + k)), f (i + k)) \<in> S"
matT_vec_multI: fixes nr :: "nat"   and nc :: "nat"   and m :: "'c list list"   and ze :: "'c"   and pl :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and ti :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and v :: "'c list" assumes "mat nr nc m" shows "vec nc (matT_vec_multI ze pl ti m v)"
distrib: fixes g :: "'d \<Rightarrow> 'c"   and h :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "(\<Prod>x\<in>A. g x * h x) = prod g A * prod h A"
transfer: shows "pcr_complex_homo 1\<^sub>h\<^sub>c 1\<^sub>h"
pseudonatural_transformation_axioms: shows "pseudonatural_transformation V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1"
larger_is_join1: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a"   and x :: "'a" assumes "class.cpo below"   and "below y x" shows "cpo.join below x y = x"