hcju/isabelleps huggingface dataset:
dickson_less_pI: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'f"   and d :: "'d \<Rightarrow> nat"   and m :: "nat"   and q :: "'c \<Rightarrow>\<^sub>0 'f" assumes "gd_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "p \<in> gd_term.dgrad_p_set pair_of_term d m"   and "q \<in> gd_term.dgrad_p_set pair_of_term d m"   and "ordered_term.ord_strict_p ord_term_strict p q" shows "gd_term.dickson_less_p pair_of_term ord_term_strict d m p q"
S_M_selects_neg_lits: fixes S :: "'a literal multiset \<Rightarrow> 'a literal multiset"   and L :: "'a literal"   and M :: "'a literal multiset set"   and C :: "'a literal multiset" assumes "selection S"   and "L \<in># S_M S M C" shows "is_neg L"
fold_insert_idem: fixes A :: "'a set"   and z :: "'a"   and x :: "'a" assumes "finite A" shows "Finite_Set.fold sup z (insert x A) = sup x (Finite_Set.fold sup z A)"
hom_zero: fixes c :: "'d" shows "c * (0::'d) = (0::'d)"
assms: shows "range (snd \<circ> r) \<subseteq> sa.V"
xx_def: shows "xx = monomial (1::'a) (monomial (Suc 0) x)"
r_Linear_order: fixes r :: "('d \<times> 'd) set" assumes "MaxR r" shows "Linear_order r"
cong: fixes g :: "'a \<Rightarrow> 'b" assumes "fun_eq_on g f (\<Union> X)" shows "ChamberComplexMorphism X Y g"
ereal_inverse_negative_iff_nonpos_not_infinity': fixes x :: "ereal" shows "(inverse x < 0) = (x < 0 \<and> x \<noteq> - \<infinity>)"
simps_2: fixes c :: "float interval"   and cs :: "float interval list" shows "tmf_polys (c # cs) = (let (pf, pi) = tmf_polys cs in (CN (mid c)\<^sub>p 0 pf, CN (centered c)\<^sub>p 0 pi))"
lists_succ_snoc: fixes xss :: "'b list list"   and xs :: "'b list" shows "lists_succ (xss @ [xs]) = lists_succ xss \<circ> list_succ xs"
linear_surj_right_inverse: fixes f :: "'b \<Rightarrow> 'c"   and T :: "'c set"   and S :: "'b set" assumes "linear (*a) (*b) f"   and "vs2.span T \<subseteq> f ` vs1.span S" shows "\<exists>g. range g \<subseteq> vs1.span S \<and> linear (*b) (*a) g \<and> (\<forall>x\<in>vs2.span T. f (g x) = x)"
delete_edges_edges_mono: fixes E' :: "('b \<times> 'b) set"   and E :: "('b \<times> 'b) set"   and G :: "'b graph" assumes "E' \<subseteq> E" shows "edges (delete_edges G E) \<subseteq> edges (delete_edges G E')"
dom: shows "dom (map_of pre) \<inter> dom (map_of FDTs) = {}"
G_is_full: shows "full_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F"
INF_inf_const1: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and 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 bot :: "'d"   and top :: "'d"   and I :: "'e set"   and x :: "'d"   and f :: "'e \<Rightarrow> 'd" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "I \<noteq> {}" shows "Inf ((\<lambda>i. inf x (f i)) ` I) = inf x (Inf (f ` I))"
EXcrypt_def: fixes K :: "nat"   and EK :: "nat"   and M :: "msg"   and m :: "msg" shows "EXcrypt K EK M m = \<lbrace>Crypt K \<lbrace>M, Hash m\<rbrace>, Crypt EK \<lbrace>Key K, m\<rbrace>\<rbrace>"
f_def: fixes env_es_s :: "v sem_env \<times> exp0 list \<times> 'a state" shows "f env_es_s = (case env_es_s of (env, es, s) \<Rightarrow> evaluate_list True env s es)"
coinduct_strong: fixes R :: "'b tree \<Rightarrow> 'b tree \<Rightarrow> bool"   and tree :: "'b tree"   and tree' :: "'b tree" assumes "R tree tree'"   and "\<And>tree tree'. R tree tree' \<Longrightarrow> root tree = root tree' \<and> (R (left tree) (left tree') \<or> left tree = left tree') \<and> (R (right tree) (right tree') \<or> right tree = right tree')" shows "tree = tree'"
exec_always: fixes s :: "stmt"   and \<sigma> :: "char list \<Rightarrow> nat option" shows "\<exists>\<rho>. s \<rhd> \<sigma> \<leadsto> \<rho>"
IH: fixes A :: "real mpoly list"   and L' :: "atom list" assumes "find_eq var L = (A, L')" shows "\<forall>p\<in>set A. MPoly_Type.degree p var = 1 \<or> MPoly_Type.degree p var = 2"
step_3_c_inner_loop_dims_main: fixes val :: "'b"   and l :: "nat"   and i :: "nat"   and j :: "nat"   and A :: "'b mat" shows "dim_row (step_3_c_inner_loop val l i j A) = dim_row A \<and> dim_col (step_3_c_inner_loop val l i j A) = dim_col A"
fp_ub_def: shows "fp_ub = (\<lambda>p. real_of_int (\<bar>int c\<bar> ^ (r * p - 1)) / fact (p - 1) * (Radius ^ (p - 1) * Q_ub ^ p))"
multiple_proper_design: fixes n :: "nat" assumes "0 < n" shows "proper_design \<V> (des.multiple_blocks n)"
is_natural_transformation: shows "natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) F H local.map"
is_nth_power_upto_unit_mult_coprimeD1: fixes x :: "'a"   and y :: "'a"   and n :: "nat" assumes "coprime x y"   and "is_nth_power_upto_unit n (x * y)" shows "is_nth_power_upto_unit n x"
indRelLTPO_impl_TRel_respects_barbs: fixes TRel :: "('procT \<times> 'procT) set" assumes "rel_respects_binary_pred (indRelLTPO TRel) (\<lambda>a. hasBarb a (STCalWB SWB TWB))" shows "rel_respects_binary_pred TRel (\<lambda>a. hasBarb a TWB)"
can_swap_neighboring_pre_and_postrecording_events: fixes i :: "nat"   and j :: "nat"   and t :: "('a, 'b, 'c) event list"   and p :: "'a"   and q :: "'a" assumes "i < j"   and "j < length t"   and "occurs_on (t ! i) = p"   and "occurs_on (t ! j) = q"   and "\<forall>k. i < k \<and> k < j \<longrightarrow> \<not> regular_event (t ! k)"   and "postrecording_event t i"   and "prerecording_event t j"   and "trace init t final" shows "can_occur (t ! j) (S t i)"
denom_def: shows "denom = snd (quot_of_fract x) * snd (quot_of_fract y)"
assms_4: shows "fixespointwise (f \<circ> g) Y"
\<chi>_def: shows "\<chi> = (\<lambda>i. f)"
Progress: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and E :: "char list \<Rightarrow> ty option"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e :: "char list exp"   and T :: "ty" assumes "wf_J_prog P"   and "P,E,s \<turnstile> e : T \<surd>"   and "\<D> e \<lfloor>dom (lcl s)\<rfloor>"   and "\<not> final e" shows "\<exists>e' s'. P \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>"
len: shows "length (row (mat_multI ze pl ti nr m1 m2) i) = length (map (\<lambda>j. scalar_prodI ze pl ti (row m1 i) (col m2 j)) [0..<nc])"
first_t_fusion: fixes x :: "'a \<times> ('b \<times> 'a) list"   and y :: "'a \<times> ('b \<times> 'a) list" assumes "Dioid_Models.last x = first y" shows "first (t_fusion x y) = first x"
ln_2_ge_56_81: shows "56 / 81 \<le> ln 2"
simps_2: fixes v :: "'a" shows "Some v \<nabla> None = Some v"
sign_comp: fixes l\<^sub>1 :: "'a literal"   and l\<^sub>2 :: "'a literal" shows "(sign l\<^sub>1 \<noteq> sign l\<^sub>2 \<and> get_pred l\<^sub>1 = get_pred l\<^sub>2 \<and> get_terms l\<^sub>1 = get_terms l\<^sub>2) = (l\<^sub>2 = l\<^sub>1\<^sup>c)"
add_zeror: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and zero :: "'a"   and x :: "'a" assumes "class.ab_near_semiring_one_zerol plus times one zero" shows "plus x zero = x"
Infinitesimal_add_approx_self2: fixes y :: "'a star"   and x :: "'a star" assumes "y \<in> Infinitesimal" shows "x \<approx> y + x"
cobounded1: fixes a :: "'a"   and b :: "'a" shows "min a b \<le> a"
simps: fixes a1 :: "'a \<times> ('a, 'b) operation"   and a2 :: "'a \<times> ('a, 'b) operation" shows "hb a1 a2 = ((\<exists>m1 i m2. a1 = m1 \<and> a2 = m2 \<and> Broadcast m1 \<sqsubset>\<^sup>i Broadcast m2) \<or> (\<exists>m1 i m2. a1 = m1 \<and> a2 = m2 \<and> Deliver m1 \<sqsubset>\<^sup>i Broadcast m2) \<or> (\<exists>m1 m2 m3. a1 = m1 \<and> a2 = m3 \<and> hb m1 m2 \<and> hb m2 m3))"
inject_2: fixes x2 :: "'c error_result"   and y2 :: "'c error_result" shows "(Rerr x2 = Rerr y2) = (x2 = y2)"
simps: fixes fs :: "'b list"   and rs :: "'b list" shows "deq (fs, rs) = (if fs = [] then (fs, rs) else norm (tl fs, rs))"
assms_3: shows "Per A B C \<or> Obtuse A B C"
filt_equiv_measurable: fixes F :: "'c \<Rightarrow> 'd measure"   and M :: "'d measure"   and N :: "'d measure"   and f :: "'d \<Rightarrow> 'e"   and P :: "'e measure" assumes "filt_equiv F M N"   and "f \<in> M \<rightarrow>\<^sub>M P" shows "f \<in> N \<rightarrow>\<^sub>M P"
eq_refl: fixes x :: "'c multiset"   and y :: "'c multiset" assumes "x = y" shows "x \<subseteq># y"
underS_def: fixes r :: "('a \<times> 'a) set"   and a :: "'a" shows "underS r a \<equiv> {b. b \<noteq> a \<and> (b, a) \<in> r}"
divide_out_code: fixes p :: "'a"   and x :: "'a" shows "divide_out p x = divide_out_aux p (x, 0)"
ar_coset_same2: fixes I :: "'a set"   and a :: "'a"   and b :: "'a" assumes "ideal R I"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "a \<uplus>\<^bsub>R\<^esub> I = b \<uplus>\<^bsub>R\<^esub> I" shows "b \<plusminus> -\<^sub>a a \<in> I"
iso_def: fixes f :: "'a" shows "S.iso f \<equiv> \<exists>g. S.inverse_arrows f g"
sorted_sort: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and xs :: "'g list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (linorder.sort_key less_eq (\<lambda>x. x) xs)"
psi_def: fixes p :: "real \<times> real"   and q :: "real \<times> real"   and r :: "real \<times> real" shows "psi p q r = (Counterclockwise_2D_Arbitrary.lex p q \<and> Counterclockwise_2D_Arbitrary.lex q r)"
Suc: shows "wlp_while_n (M 0) (M 1) WS k P \<le>\<^sub>L wlp_while_n (M 0) (M 1) WS k Q"
acomplete_subtreeL: fixes l :: "'c tree"   and x :: "'c"   and r :: "'c tree" assumes "acomplete \<langle>l, x, r\<rangle>" shows "acomplete l"
hd_stack_in_tj_stack: assumes "stack s \<noteq> []" shows "hd (stack s) \<in> set (tj_stack s)"
divide_eq_eq_numeral1_1: fixes b :: "'b"   and w :: "num"   and a :: "'b" shows "(b / numeral w = a) = (if numeral w \<noteq> (0::'b) then b = a * numeral w else a = (0::'b))"
V: shows "V = zip (map getOneIp (build_ip_partition c rs)) (build_ip_partition c rs)"
max_n_in_n: shows "Max_mset N \<in># N"
permutation_of_list_def: fixes xs :: "('b \<times> 'b) list"   and x :: "'b" shows "permutation_of_list xs x = (case map_of xs x of None \<Rightarrow> x | Some y \<Rightarrow> y)"
remainder_conj: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a"   and \<psi> :: "'a" assumes "K = Cn A"   and "K \<turnstile> \<phi> .\<and>. \<psi>" shows "K .\<bottom>. (\<phi> .\<and>. \<psi>) = (K .\<bottom>. \<phi>) \<union> (K .\<bottom>. \<psi>)"
rep_eq: shows "pdevs_apply zero_pdevs = (\<lambda>_. 0::'b)"
span_finite_subset: fixes p :: "'b"   and B :: "'b set" assumes "p \<in> ideal B" obtains A :: "'b set" where "finite A" and "A \<subseteq> B" and "p \<in> ideal A"
esi_def: shows "esi = (\<lambda>j. local.xij_f (Suc i) (Suc i + j) * local.pprod k (Suc i) (Suc i + j))"
dropWhile_eq_obtain_leading: fixes P :: "'f \<Rightarrow> bool"   and xs :: "'f list"   and ys :: "'f list" assumes "dropWhile P xs = ys" obtains zs :: "'f list" where "xs = zs @ ys" and "\<And>z. z \<in> set zs \<Longrightarrow> P z" and "no_leading P ys"
Unifier_Idem_subst: fixes r :: "('b \<times> 'b trm) list"   and s :: "('b \<times> 'b trm) list"   and t :: "'b trm"   and u :: "'b trm" assumes "Idem r"   and "Unifier s (t \<lhd> r) (u \<lhd> r)" shows "Unifier (r \<lozenge> s) (t \<lhd> r) (u \<lhd> r)"
higherPowers_lowerPowers: fixes n :: "nat"   and i :: "nat"   and x :: "nat \<Rightarrow>\<^sub>0 'b" shows "higherPowers n i (lowerPowers n i x) = x"
latestActNxtAct: fixes n :: "nat"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and tid :: "'a"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>n'\<ge>n. active tid (t n')"   and "\<exists>n'<n. active tid (t n')" shows "dynamic_component.latestAct active tid t n < dynamic_component.nxtAct active tid t n"
nth_butlast: fixes n :: "nat"   and xs :: "'a list" assumes "n < length (butlast xs)" shows "butlast xs ! n = xs ! n"
subtype_reflexivity_from_typing: fixes \<Gamma> :: "binding list"   and t :: "trm"   and T :: "ty" assumes "\<Gamma> \<turnstile> t : T" shows "\<Gamma>\<turnstile>T<:T"
Min_le: fixes A :: "('x \<Rightarrow>\<^sub>0 nat) set"   and x :: "'x \<Rightarrow>\<^sub>0 nat" assumes "finite A"   and "x \<in> A" shows "ordered_powerprod_lin.Min A \<preceq> x"
splitset: shows "{0..<Suc k} \<union> {Suc (Suc k)..<n} = vars1 - {Suc k}"
augment_mono_map': fixes A :: "nat \<Rightarrow> bool option"   and B :: "nat \<Rightarrow> bool option"   and x :: "nat set"   and C :: "nat set" assumes "A \<subseteq>\<^sub>m B" shows "A \<oplus>\<^bsub>x\<^esub> C \<subseteq>\<^sub>m B \<oplus>\<^bsub>x\<^esub> C"
cf: shows "coeffs f = map (poly.coeff f) [0..<Suc (degree f)]"
N: shows "N \<in> M \<rightarrow>\<^sub>M subprob_algebra S"
sas_plus_equivalent_to_strips_i_a_X: fixes s :: "'c \<Rightarrow> 'd option"   and V :: "'c set"   and D :: "'c \<Rightarrow> 'd list option"   and ops :: "('c, 'd) sas_plus_operator list" assumes "dom s \<subseteq> V"   and "V \<subseteq> dom D"   and "\<forall>v\<in>dom s. the (s v) \<in> set (the (D v))"   and "\<forall>op\<in>set ops. \<forall>(v, a)\<in>set (effect_of op). v \<in> V \<and> a \<in> set (the (D v))" shows "\<forall>v\<in>dom (execute_parallel_operator_sas_plus s ops). the (execute_parallel_operator_sas_plus s ops v) \<in> set (the (D v))"
to_mset_def: fixes t :: "('a, 'b) BinomialHeap" shows "BinomialHeap_loc.to_mset t \<equiv> BinomialHeapStruc.queue_to_multiset (Rep_BinomialHeap t)"
pathstart_in_path_image: fixes g :: "real \<Rightarrow> 'c" shows "pathstart g \<in> path_image g"
last_coeff_is_hd: fixes xs :: "'b list" assumes "xs \<noteq> []" shows "coeff (Poly xs) (length xs - 1) = hd (rev xs)"
obtain_fresh: fixes x :: "'c" obtains a :: "'b" where "atom a \<sharp> x"
andfold_MatchExp_matches: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and ms :: "'b match_expr list"   and a :: "action"   and p :: "'c" shows "matches \<gamma> (andfold_MatchExp ms) a p = (\<forall>m\<in>set ms. matches \<gamma> m a p)"
domains_char: fixes f :: "'a" shows "C.domains f = (if C.arr f then {C.dom f} else {})"
replicate_def: shows "replicate \<equiv> rec_nat (\<lambda>x. []) (\<lambda>n na x. x # na x)"
fv_pair_fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_subset: fixes d :: "'a \<times> ('fun, 'b) Term.term \<times> ('fun, 'b) Term.term"   and D :: "('a \<times> ('fun, 'b) Term.term \<times> ('fun, 'b) Term.term) list" assumes "d \<in> set D" shows "fv (pair (snd d)) \<subseteq> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (unlabel D)"
continuous_on_inv: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a" assumes "continuous_on s f"   and "compact s"   and "\<forall>x\<in>s. g (f x) = x" shows "continuous_on (f ` s) g"
plus_gt_left_mono: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "delta_gt \<delta> x y" shows "delta_gt \<delta> (x + z) (y + z)"
assms_1: shows "\<forall>a\<in>V - {v}. \<forall>b\<in>V - {v}. nodes_connected G a b"
eq_iff: fixes x :: "'b"   and y :: "'b" shows "(x = y) = (ord' x y \<and> ord' y x)"
reds_preserves_bconf: fixes es :: "char list exp list"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool"   and es' :: "char list exp list"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "P \<turnstile> \<langle>es,(h, l, sh),b\<rangle> [\<rightarrow>] \<langle>es',(h', l', sh'),b'\<rangle>"   and "iconfs sh es"   and "P,sh \<turnstile>\<^sub>b (es,b) \<surd>" shows "P,sh' \<turnstile>\<^sub>b (es',b') \<surd>"
tendsto_compose: fixes g :: "'c \<Rightarrow> 'd"   and l :: "'c"   and f :: "'e \<Rightarrow> 'c"   and F :: "'e filter" assumes "g \<midarrow>l\<rightarrow> g l"   and "(f \<longlongrightarrow> l) F" shows "((\<lambda>x. g (f x)) \<longlongrightarrow> g l) F"
rep_eq: fixes x :: "('a, 'b) ffun"   and xa :: "'b set" shows "pfun_of (x \<rhd>\<^sub>f xa) = pfun_of x \<rhd>\<^sub>p xa"
s_1: shows "s \<longlonglongrightarrow>\<^bsub>\<^esub> \<infinity>"
LBex: fixes f :: "atom fm"   and x :: "'a"   and xs :: "'a list" assumes "nqfree f"   and "DLO.I f (x # xs)"   and "\<not> DLO.I (inf\<^sub>- f) xs"   and "x \<notin> EQ f xs" shows "\<exists>l\<in>LB f xs. l < x"
cod_simps_2: shows "\<rho>.cod.leg0 = \<rho>.leg0"
zmult_zle_mono: fixes i :: "int"   and j :: "int"   and k :: "int" assumes "i \<le> j"   and "0 < k" shows "k * i \<le> k * j"
SUP_eqI: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'b"   and x :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq (f i) x"   and "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> less_eq (f i) y) \<Longrightarrow> less_eq x y" shows "Sup (f ` A) = x"
g_1: shows "(g \<longlongrightarrow> 0) F"
intro: fixes s1 :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and B1 :: "'d set"   and s2 :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and B2 :: "'e set" assumes "finite_dimensional_vector_space s1 B1"   and "finite_dimensional_vector_space s2 B2" shows "finite_dimensional_vector_space_pair s1 B1 s2 B2"
unfold_2: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b" shows "(\<diamond>P) \<sigma> = (P \<sigma> \<or> (\<circle>\<diamond>P) \<sigma>)"
s_t_stop: shows "delayed_safe_distance.other.s_stop = delayed_safe_distance.other.p_max"
assms_3: shows "i \<noteq> 0 \<or> j \<noteq> 0"
decomp_propa: fixes qs :: "(nat \<times> 'b) list"   and ss :: "'b list"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and w :: "nat set" assumes "\<forall>(q, t)\<in>set qs. q < length ss" shows "propa f qs ss w = (merges f qs ss, {q. \<exists>t. (q, t) \<in> set qs \<and> t \<squnion>\<^bsub>f\<^esub> ss ! q \<noteq> ss ! q} \<union> w)"
restrict_group_imp_subgroup: fixes H :: "'a set" assumes "H \<subseteq> carrier G"   and "Group.group (G\<lparr>carrier := H\<rparr>)" shows "subgroup H G"
ZObis_sym: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and c :: "('c, 'd, 'a) cmd"   and d :: "('c, 'd, 'a) cmd"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool" assumes "PL_Indis cval indis"   and "(c, d) \<in> PL_Indis.ZObis aval tval cval indis" shows "(d, c) \<in> PL_Indis.ZObis aval tval cval indis"
simps_1: fixes g1 :: "('a \<Rightarrow> bool) \<times> ('a \<Rightarrow> 'b \<times> 'a)"   and g2 :: "('c \<Rightarrow> bool) \<times> ('c \<Rightarrow> 'b \<times> 'c)"   and s1 :: "'a"   and s2 :: "'c" shows "append_has_next g1 g2 (Inl (s1, s2)) = (fst g1 s1 \<or> fst g2 s2)"
cone_def: fixes J :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and C :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and D :: "'ag \<Rightarrow> 'ah"   and a :: "'ah"   and \<chi> :: "'ag \<Rightarrow> 'ah" shows "Limit.cone J C D a \<chi> \<equiv> (category C \<and> category J) \<and> diagram J C D \<and> constant_functor J C a \<and> natural_transformation J C (constant_functor.map J C a) D \<chi>"
prems_3: shows "InvariantCEntailed (getConflictFlag state') F0 (getC state')"
double_negation: fixes P :: "'b \<Rightarrow> bool" shows "(\<not> \<not> P) = P"
integrable_integral: fixes f :: "'c \<Rightarrow> 'd"   and i :: "'c set" assumes "f integrable_on i" shows "(f has_integral integral i f) i"
meet_equality: fixes x :: "'c"   and y :: "'c"   and inf :: "'c" assumes "is_inf x y inf" shows "x \<sqinter> y = inf"
step_preserves_SN_on: fixes a :: "'b"   and b :: "'b"   and r :: "('b \<times> 'b) set" assumes "(a, b) \<in> r"   and "SN_on r {a}" shows "SN_on r {b}"
select_convs_1: fixes Field1 :: "nat message_af"   and Field2 :: "int message_af"   and Field3 :: "int message_af"   and more :: "'b" shows "Field1 \<lparr>Field1 = Field1, Field2 = Field2, Field3 = Field3, \<dots> = more\<rparr> = Field1"
Vagree_repv_self: fixes \<omega> :: "variable \<Rightarrow> real"   and x :: "variable"   and d :: "real" shows "Vagree \<omega> (repv \<omega> x d) {x} = (d = \<omega> x)"
antipar_2: shows "src (tab\<^sub>0 f)\<^sup>* = trg (tab\<^sub>0 f)"
ordertype_le_ordertype: fixes r :: "('a \<times> 'a) set"   and A :: "'a set"   and s :: "('b \<times> 'b) set"   and B :: "'b set" assumes "wf r"   and "total_on A r"   and "small A"   and "wf s"   and "total_on B s"   and "trans s"   and "small B" shows "(ordertype A r \<le> ordertype B s) = (\<exists>f\<in>A \<rightarrow> B. inj_on f A \<and> (\<forall>x\<in>A. \<forall>y\<in>A. (x, y) \<in> r \<longrightarrow> (f x, f y) \<in> s))"
B3: shows "B3 = 0\<^sub>m (n - k) k"
abs_eq: fixes x :: "complex" shows "of_complex x = abs_complex_homo (of_complex_hcoords x)"
Cons_x_xs: fixes ys :: "'d list" shows "zip (map f xs) (map g ys) = map2 (\<lambda>x y. (f x, g y)) xs ys"
object_ptr_kinds_M_eq_h2: fixes ptrs :: "(_) object_ptr list" shows "h2 \<turnstile> object_ptr_kinds_M \<rightarrow>\<^sub>r ptrs = h3 \<turnstile> object_ptr_kinds_M \<rightarrow>\<^sub>r ptrs"
map_of_zip_map: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> 'b" shows "map_of (zip xs (map f xs)) = (\<lambda>x. if x \<in> set xs then Some (f x) else None)"
runit_simps_5: fixes f :: "('a S.E.term, 'a) S.arr" assumes "S.ide f" shows "S.cod (S.runit f) = f"
set2_F_bound: fixes x :: "('v, 'n, 'w, 'o, 'p, 'q, 'r, 's, 't, 'u) F" shows "|set2_F x| \<le>o bd_F"
refine: shows "(succ_of_list_impl, succ_of_list) \<in> \<langle>nat_rel \<times>\<^sub>r nat_rel\<rangle>list_rel \<rightarrow> nat_rel \<rightarrow> \<langle>nat_rel\<rangle>list_set_rel"
lfinite_conv_llength_enat: fixes xs :: "'a llist" shows "lfinite xs = (\<exists>n. llength xs = enat n)"
Sigma_mono: fixes A :: "'a set"   and C :: "'a set"   and B :: "'a \<Rightarrow> 'b set"   and D :: "'a \<Rightarrow> 'b set" assumes "A \<subseteq> C"   and "\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> D x" shows "Sigma A B \<subseteq> Sigma C D"
infinite: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "infinite A" shows "aggregation.sum_0 g A = zero"
linear_span_sub: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and A :: "'f set"   and H :: "'e set" assumes "R module M"   and "ideal R A"   and "H \<subseteq> carrier M" shows "linear_span R M A H \<subseteq> carrier M"
dfa_trans_is_node: fixes A :: "nat bdd list \<times> bool list"   and n :: "nat"   and q :: "nat"   and a :: "bool list" assumes "wf_dfa A n"   and "dfa_is_node A q"   and "is_alph n a" shows "dfa_is_node A (dfa_trans A q a)"
rel_pred_def: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_pred R = R ===> (=)"
plus_complex_interval: fixes x :: "complex"   and X :: "complex_interval"   and y :: "complex"   and Y :: "complex_interval" assumes "x \<in>\<^sub>c X"   and "y \<in>\<^sub>c Y" shows "x + y \<in>\<^sub>c X + Y"
in_tset_conv_tnth: fixes x :: "'g"   and xs :: "('g, 'h) tllist" shows "(x \<in> tset xs) = (\<exists>n. enat n < tlength xs \<and> tnth xs n = x)"
wfp_on_mulex1_multisets: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "wfp_on P A" shows "wfp_on (mulex1 P) (multisets A)"
proj_Cons_3: fixes m :: "'d"   and n :: "'d"   and a :: "'e"   and A :: "('d strand_label \<times> 'e) list" assumes "m \<noteq> n" shows "proj n ((ln m, a) # A) = proj n A"
int_simps_15: fixes P :: "'b \<Rightarrow> bool" shows "\<turnstile> (P \<longrightarrow> \<not> P) = (\<not> P)"