hcju/isabelleps huggingface dataset:
vsubstEnv_def2: fixes ys :: "'varSort"   and y1 :: "'var"   and y :: "'var"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option" shows "rho &[y1 // y]_ys = (\<lambda>xs x. case rho xs x of None \<Rightarrow> if xs = ys \<and> x = y then Some (Var ys y1) else None | Some X \<Rightarrow> Some (X #[y1 // y]_ys))"
self_dvd_gauss_cnj_iff: fixes z :: "gauss_int" shows "(z dvd gauss_cnj z) = (ReZ z = 0 \<or> ImZ z = 0 \<or> \<bar>ReZ z\<bar> = \<bar>ImZ z\<bar>)"
real_interval_minus: fixes a :: "float interval"   and b :: "float interval" shows "real_interval (a - b) = real_interval a - real_interval b"
add_mset_add_mset_same_iff: fixes a :: "'d"   and A :: "'d multiset"   and B :: "'d multiset" shows "(add_mset a A = add_mset a B) = (A = B)"
undiscr_Discr: fixes x :: "'a" shows "undiscr (Discr x) = x"
simps_2: fixes x :: "'b memref"   and xs :: "'b memref list"   and S :: "nat \<Rightarrow> bool option" shows "share (x # xs) S = (case x of Write\<^sub>s\<^sub>b volatile x xa xb A L R W \<Rightarrow> if volatile then share xs (S \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L) else share xs S | Ghost\<^sub>s\<^sub>b A L R W \<Rightarrow> share xs (S \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L) | _ \<Rightarrow> share xs S)"
a'_3: shows "T_splay a' t = T_splay a t"
factorization_m_def: fixes m :: "int"   and f :: "int poly"   and cfs :: "int \<times> int poly multiset" shows "poly_mod.factorization_m m f cfs \<equiv> case cfs of (c, fs) \<Rightarrow> poly_mod.Mp m f = poly_mod.Mp m (smult c (\<Prod>\<^sub># fs)) \<and> (\<forall>f\<in>#fs. poly_mod.irreducible\<^sub>d_m m f \<and> monic (poly_mod.Mp m f))"
is_nonterminal_startsymbol: shows "is_nonterminal \<SS>"
col_A_B_lAB: fixes A :: "Points"   and B :: "Points"   and l :: "Lines" shows "col A B (Pascal_Property.inter l (line A B))"
well_def_wlp_nearly_healthy: fixes prog :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "well_def prog" shows "nearly_healthy (wlp prog)"
val_by_cf: shows "f.val = (\<Sum>(u, v)\<in>outgoing s. f.cf (v, u))"
infinite_lessThan_int: fixes n :: "int" shows "infinite {..<n}"
wtl_Cons: fixes i :: "'b"   and "is" :: "'b list"   and c :: "'a list"   and pc :: "nat"   and s :: "'a" shows "wtl (i # is) c pc s = (let s' = wtc c pc s in if s' = \<top> \<or> s = \<top> then \<top> else wtl is c (pc + 1) s')"
set_mset_single: fixes b :: "'b" shows "set_mset (b, {#}) = {b}"
L_mono: fixes S1 :: "nat rexp set"   and S2 :: "nat rexp set" assumes "S1 \<subseteq> S2" shows "L S1 \<subseteq> L S2"
size_filter_neg: fixes P :: "'b \<Rightarrow> bool"   and A :: "'b multiset" shows "size (filter_mset P A) = size A - size {#a \<in># A. \<not> P a#}"
take_takefill': fixes n :: "nat"   and k :: "nat"   and m :: "nat"   and fill :: "'b"   and w :: "'b list" assumes "n = k + m" shows "take k (takefill fill n w) = takefill fill k w"
vector_derivative_chain_at: fixes f :: "real \<Rightarrow> real"   and x :: "real"   and g :: "real \<Rightarrow> 'a" assumes "f differentiable at x"   and "g differentiable at (f x)" shows "vector_derivative (g \<circ> f) (at x) = vector_derivative f (at x) *\<^sub>R vector_derivative g (at (f x))"
cong: fixes A :: "'b set"   and B :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> g x = h x" shows "sum g A = sum h B"
simps_1: fixes x :: "'v" shows "wt0 (Var x) = PVar (PWt x)"
space_empty_iff: fixes N :: "'c measure" shows "(space N = {}) = (sets N = {{}})"
image_subset_iff_funcset: fixes F :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and B :: "'b set" shows "(F ` A \<subseteq> B) = (F \<in> A \<rightarrow> B)"
vfsequence_axioms: shows "vfsequence xs\<^sub>1"
MGU_Var2: fixes v :: "'a"   and t :: "('b, 'a) Term.term" assumes "v \<notin> fv t" shows "MGU (Var(v := t)) (Var v) t"
C_disj: fixes i :: "nat"   and j :: "nat"   and M2 :: "('e, 'f, 'g) FSM"   and M1 :: "('e, 'f, 'h) FSM"   and \<Omega> :: "('e, 'f) ATC set"   and V :: "'e list set"   and m :: "nat" assumes "i \<noteq> j" shows "C M2 M1 \<Omega> V m i \<inter> C M2 M1 \<Omega> V m j = {}"
that_2: fixes x :: "'a" shows "(f x = (0::'b)) = (x = (0::'a))"
mult_1_left: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.monoid_mult one times" shows "times one a = a"
subst_eq_if_eq_vars: fixes \<theta> :: "'c \<Rightarrow> ('d, 'e) Term.term"   and \<sigma> :: "'c \<Rightarrow> ('d, 'e) Term.term" assumes "\<And>v. Var v \<cdot> \<theta> = Var v \<cdot> \<sigma>" shows "\<theta> = \<sigma>"
is_expansion_aux_basis_nonempty: fixes F :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list" assumes "is_expansion_aux F f basis" shows "basis \<noteq> []"
iso_iff_group_isomorphisms: fixes f :: "('a \<Rightarrow>\<^sub>0 int) \<Rightarrow> 'c"   and H :: "('c, 'd) monoid_scheme" shows "(f \<in> Group.iso (free_Abelian_group S) H) = (\<exists>g. group_isomorphisms (free_Abelian_group S) H f g)"
set_cases_3: fixes e :: "'g"   and a :: "('h, 'i, 'g) generat" assumes "e \<in> generat_conts a" obtains z1 :: "'i" where "a = IO z1 e"
plus_one_helper: fixes x :: "'b word"   and n :: "'b word" assumes "x < n + 1" shows "x \<le> n"
apath_nonempty_ends: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "G.apath u p v"   and "p \<noteq> []" shows "u \<noteq> v"
u: fixes g :: "'a \<Rightarrow>f 'b"   and a :: "'a"   and b :: "'b" shows "f g(a $:= b) = upd g a b (f g)"
WbisT_matchC_MC_rev: fixes s :: "('test, 'atom) com"   and t :: "('test, 'atom) com" assumes "s \<approx>wT t" shows "matchC_MC WbisT t s"
antisym_conv2: fixes x :: "'a list"   and y :: "'a list" assumes "suffix x y" shows "(\<not> strict_suffix x y) = (x = y)"
mid_centered_collapse: fixes abs_bound :: "float interval" shows "interval_of (real_of_float (mid abs_bound)) + real_interval (centered abs_bound) = real_interval abs_bound"
vars\<^sub>s\<^sub>s\<^sub>t_def: fixes S :: "('d, 'c) stateful_strand_step list" shows "vars\<^sub>s\<^sub>s\<^sub>t S \<equiv> \<Union> (set (map vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p S))"
distinct_11: fixes x21 :: "variable"   and x22 :: "typ"   and x41 :: "typ"   and x42 :: "Core.term" shows "Fv x21 x22 \<noteq> Abs x41 x42"
valid_cfg_state_in_S: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and cfg :: "'a cfg" assumes "Markov_Decision_Process_Invariant K S"   and "cfg \<in> Markov_Decision_Process_Invariant.valid_cfg K S" shows "state cfg \<in> S"
rCase: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and B :: "('a, 'b, 'c) boundOutput"   and \<phi> :: "'c"   and Cs :: "('c \<times> ('a, 'b, 'c) psi) list"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> ROut M B"   and "\<And>C. Prop C \<Psi> P M B"   and "(\<phi>, P) mem Cs"   and "\<Psi> \<turnstile> \<phi>"   and "guarded P" shows "Prop C \<Psi> (Cases Cs) M B"
supp_Some: fixes x :: "'b" shows "supp (Some x) = supp x"
\<beta>\<^sub>m\<^sub>a\<^sub>g\<^sub>i\<^sub>c_approximating_bigstep_fun_iff_iptables_bigstep: fixes rs :: "'c rule list"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and \<alpha> :: "action \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and s :: "state"   and t :: "state"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "good_ruleset rs" shows "(approximating_bigstep_fun (\<beta>\<^sub>m\<^sub>a\<^sub>g\<^sub>i\<^sub>c \<gamma>, \<alpha>) p rs s = t) = \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t"
stutter_extend_def: fixes G :: "('c, 'd) graph_rec_scheme" shows "stutter_extend G \<equiv> \<lparr>g_V = g_V G, g_E = stutter_extend_edges (g_V G) (g_E G), g_V0 = g_V0 G, \<dots> = graph_rec.more G\<rparr>"
sym_group_inv_closed: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "p \<in> carrier (sym_group n)" shows "inv' p \<in> carrier (sym_group n)"
hd_in_set: fixes xs :: "'e list" assumes "xs \<noteq> []" shows "hd xs \<in> set xs"
eq_refl: fixes x :: "'a"   and y :: "'a" assumes "x = y" shows "less_eq0 x y"
ide_char: fixes a :: "'a" shows "L'.C.ide a = (L'.C.arr a \<and> L'.C.dom a = a \<and> L'.C.cod a = a)"
R0: fixes X :: "('c, 'd) ipurge_rec" shows "X \<in> ipurge_tr_rev_t_set X"
lspasl_magicl: fixes Gamma :: "bool"   and h1 :: "'a"   and h2 :: "'a"   and h0 :: "'a"   and A :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> bool"   and Delta :: "bool" assumes "Gamma \<and> (h1,h2\<triangleright>h0) \<and> (A \<longrightarrow>* B) h2 \<longrightarrow> A h1 \<or> Delta"   and "Gamma \<and> (h1,h2\<triangleright>h0) \<and> (A \<longrightarrow>* B) h2 \<and> B h0 \<longrightarrow> Delta" shows "Gamma \<and> (h1,h2\<triangleright>h0) \<and> (A \<longrightarrow>* B) h2 \<longrightarrow> Delta"
mult_eq_0_iff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.semiring_no_zero_divisors plus zero times" shows "(times a b = zero) = (a = zero \<or> b = zero)"
addrow_mat_carrier: fixes n :: "nat"   and a :: "'b"   and k :: "nat"   and l :: "nat" shows "addrow_mat n a k l \<in> carrier_mat n n"
assms: shows "timpls_transformable_to_pred A (Fun f T) t"
in_lex_prod: fixes a :: "'a"   and b :: "'c"   and a' :: "'a"   and b' :: "'c"   and r :: "('a \<times> 'a) set"   and s :: "('c \<times> 'c) set" shows "(((a, b), a', b') \<in> r <*lex*> s) = ((a, a') \<in> r \<or> a = a' \<and> (b, b') \<in> s)"
sconf_def: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" shows "P,E \<turnstile> s \<surd> \<equiv> let (h, l) = s in P \<turnstile> h \<surd> \<and> P,h \<turnstile> l (:\<le>)\<^sub>w E \<and> P \<turnstile> E \<surd>"
PLS_nempty_and_has_min: fixes s :: "('c, 'd) fmap"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" shows "\<exists>x. x \<in> PLS s as \<and> x = Inf (PLS s as)"
valid_indexE: fixes "is" :: "nat list"   and ds :: "nat list" assumes "is \<lhd> ds" obtains "is = []" and "ds = []"   | isa :: "nat list" and dsa :: "nat list" and i :: "nat" and d :: "nat" where "is = i # isa" and "ds = d # dsa" and "isa \<lhd> dsa" and "i < d"
is_map_to_sorted_list_def: fixes ordR :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and Rk :: "('g \<times> 'f) set"   and Rv :: "('h \<times> 'i) set"   and Rm :: "('g \<times> 'f) set \<Rightarrow> ('h \<times> 'i) set \<Rightarrow> ('j \<times> ('f \<Rightarrow> 'i option)) set"   and tsl :: "'j \<Rightarrow> ('g \<times> 'h) list" shows "is_map_to_sorted_list ordR Rk Rv Rm tsl \<equiv> \<forall>m m'. (m, m') \<in> \<langle>Rk, Rv\<rangle>Rm \<longrightarrow> (\<exists>l'. (tsl m, l') \<in> \<langle>Rk \<times>\<^sub>r Rv\<rangle>list_rel \<and> RETURN l' \<le> it_to_sorted_list (key_rel ordR) (map_to_set m'))"
cblinfun_of_mat_times: fixes M :: "complex mat"   and N :: "complex mat" assumes "M \<in> carrier_mat (length canonical_basis) (length canonical_basis)"   and "N \<in> carrier_mat (length canonical_basis) (length canonical_basis)" shows "cblinfun_of_mat (M * N) = cblinfun_of_mat M o\<^sub>C\<^sub>L cblinfun_of_mat N"
unfoldE_1: fixes P :: "pi"   and Q :: "pi"   and s :: "(name \<times> name) list" assumes "P \<simeq>\<^sup>s Q" shows "P[
] \<leadsto>\<guillemotleft>weakBisim\<guillemotright> Q[
]"
take_bit_not_eq_mask_diff: fixes n :: "nat"   and a :: "'a" shows "take_bit n (NOT a) = mask n - take_bit n a"
zero_less_norm_iff: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and x :: "'e" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "(0 < norm x) = (x \<noteq> zero)"
round_down_float: fixes prec :: "int"   and x :: "real" shows "round_down prec x \<in> float"
B: shows "\<forall>x\<in>set xs. key x \<in> {mi..ma}"
assms: shows "f \<approx>>\<^bsub>CatExp A B\<^esub> g"
Riemann_localization_integral_range_half: fixes f :: "real \<Rightarrow> real"   and d :: "real" assumes "f absolutely_integrable_on {- pi..pi}"   and "0 < d"   and "d \<le> pi" shows "(\<lambda>n. (LINT x|lebesgue_on {0..pi}. Dirichlet_kernel n x * (f x + f (- x))) - (LINT x|lebesgue_on {0..d}. Dirichlet_kernel n x * (f x + f (- x)))) \<longlonglongrightarrow> 0"
r2: shows "\<forall>x. x \<noteq> z \<and> dist x z < r2 \<longrightarrow> g x = f x"
ldropn_Suc_conv_ldropn: fixes n :: "nat"   and xs :: "'b llist" assumes "enat n < llength xs" shows "LCons (lnth xs n) (ldropn (Suc n) xs) = ldropn n xs"
le_less_linear: fixes x :: "'t"   and y :: "'t" shows "x \<preceq>\<^sub>t y \<or> y \<prec>\<^sub>t x"
incrIndexList_help10: fixes ls :: "nat list"   and m :: "nat"   and nmax :: "nat"   and x :: "nat" assumes "incrIndexList ls m nmax"   and "x \<in> set ls" shows "x < nmax"
simps_1: fixes l :: "'a" shows "translation (A_Lbl l) = LG {(l, 0, 1)} {0, 1}"
ran_alt_def: fixes m :: "'d \<Rightarrow> 'c option" shows "ran m = (the \<circ> m) ` dom m"
finite_acyclic_wf_converse: fixes r :: "('a \<times> 'a) set" assumes "finite r"   and "acyclic r" shows "wf (r\<inverse>)"
A5: shows "\<forall>SQ SR. SQ \<in>S Q \<and> SR \<in>S R \<longrightarrow> (\<lbrakk>SQ\<rbrakk>, \<lbrakk>SR\<rbrakk>) \<in> TRel"
net_tree_ips_commute: fixes p1 :: "net_tree"   and p2 :: "net_tree" shows "net_tree_ips (p1 \<parallel> p2) = net_tree_ips (p2 \<parallel> p1)"
bind_lunit: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'b option" shows "Option.bind \<lfloor>x\<rfloor> f = f x"
Mismatch_2: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi"   and a :: "name"   and c :: "name" assumes "P \<Longrightarrow>\<^sub>l\<^sup>^\<alpha> \<prec> P'"   and "P \<noteq> P'"   and "a \<noteq> c" shows "[a\<noteq>c]P \<Longrightarrow>\<^sub>l\<^sup>^\<alpha> \<prec> P'"
getIJProperties_6: fixes V :: "nat set"   and I :: "nat set"   and J :: "nat set"   and Q_pos :: "(nat set \<times> 'a option list set) set"   and Q_neg :: "(nat set \<times> 'a option list set) set" assumes "2 \<le> card V"   and "(I, J) = getIJ Q_pos Q_neg V" shows "I \<inter> J = {}"
simps_7: fixes z :: "octo" shows "Im6 (Octonions.cnj z) = - Im6 z"
derivable_closed: fixes x :: "'a" assumes "derivable x" shows "derivable (f x)"
remove_removes_child: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and node_ptr :: "(_) node_ptr"   and children :: "(_) node_ptr list"   and h' :: "(_) heap" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r node_ptr # children"   and "h \<turnstile> remove node_ptr \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children"
coboundedI1: fixes a :: "'c multiset"   and c :: "'c multiset"   and b :: "'c multiset" assumes "a \<subseteq># c" shows "a \<inter># b \<subseteq># c"
edg_simps_3: fixes \<Delta> :: "('a, 'b) network_scheme"   and v :: "'a antiparallel_edges.vertex"   and x :: "'a" assumes "antiparallel_edges \<Delta>" shows "antiparallel_edges.edg \<Delta> v (antiparallel_edges.Vertex x) = (\<exists>xa. v = antiparallel_edges.Edge xa x \<and> edge \<Delta> xa x)"
apd_d_def: fixes x :: "'a" shows "apd_d x = ad (ad x)"
zero: fixes a :: "real" shows "a *\<^sub>R (0::'a) = (0::'a)"
m_comm: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier (UP INTEG)"   and "y \<in> carrier (UP INTEG)" shows "x \<otimes>\<^bsub>UP INTEG\<^esub> y = y \<otimes>\<^bsub>UP INTEG\<^esub> x"
in_read_only_restrict_conv: fixes a :: "nat"   and \<S> :: "nat \<Rightarrow> bool option"   and A :: "nat set"   and L :: "nat set" shows "(a \<in> read_only (\<S> \<ominus>\<^bsub>A\<^esub> L)) = (a \<in> read_only \<S> \<and> a \<notin> L \<and> a \<notin> A)"
subst_closed_in: fixes T :: "ty"   and \<Delta> :: "binding list"   and X :: "tyvrs"   and S :: "ty"   and \<Gamma> :: "binding list"   and U :: "ty" assumes "T closed_in (\<Delta> @ TVarB X S # \<Gamma>)"   and "U closed_in \<Gamma>" shows "T[X \<mapsto> U]\<^sub>\<tau> closed_in (\<Delta>[X \<mapsto> U]\<^sub>e @ \<Gamma>)"
mono_onI: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>r s. \<lbrakk>r \<in> A; s \<in> A; r \<le> s\<rbrakk> \<Longrightarrow> f r \<le> f s" shows "mono_on f A"
control_withinD: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp"   and \<sigma> :: "('i \<times> ('f, 'g, 'e, 'h) seqp) set"   and \<xi> :: "'i"   and p :: "('f, 'g, 'e, 'h) seqp" assumes "control_within \<Gamma> \<sigma>"   and "(\<xi>, p) \<in> \<sigma>" shows "\<exists>pn. p \<in> subterms (\<Gamma> pn)"
le_supE: fixes a :: "'a"   and b :: "'a"   and x :: "'a" assumes "sup a b \<le> x" obtains "a \<le> x" and "b \<le> x"
comp_apply: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'e \<Rightarrow> 'b"   and x :: "'e" shows "(f \<circ> g) x = f (g x)"
el: shows "((idn, S), \<rho> idn S) \<in> set (map (\<lambda>a. case a of (x, y) \<Rightarrow> ((x, y), \<rho> x y)) l)"
valid_add_new_ts_mmsaux'_unfolded: fixes args :: "Monitor.args"   and cur :: "nat"   and ot :: "nat"   and gc :: "nat"   and maskL :: "bool list"   and maskR :: "bool list"   and data_prev :: "(nat \<times> 'b option list set) queue"   and data_in :: "(nat \<times> 'b option list set) queue"   and tuple_in :: "('b option list, nat) mapping"   and tuple_since :: "('b option list, nat) mapping"   and auxlist :: "(nat \<times> 'b option list set) list"   and nt :: "nat" assumes "valid_mmsaux args cur (ot, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since) auxlist"   and "cur \<le> nt" shows "valid_mmsaux args nt (add_new_ts_mmsaux' args nt (ot, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since)) auxlist"
jonsson3b: fixes x :: "'a"   and y :: "'a" assumes "\<forall>x. x \<rhd> (1::'a) \<rhd> (1::'a) = x" shows "x \<sqinter> y \<rhd> (1::'a) = (x \<rhd> (1::'a)) \<sqinter> (y \<rhd> (1::'a))"
vec_lambda_eta: fixes g :: "('d, 'e) vec" shows "(\<chi>i. g $ i) = g"
less_length_mono: fixes s :: "'a list"   and t :: "'a list" assumes "s < t" shows "length s < length t"
int_of_reals_1: fixes w :: "int"   and z :: "int" shows "of_int w + of_int z = of_int (w + z)"
rep_nat_pp_nat: shows "rep_nat_pp = (\<lambda>x. x)"
transp_on_subset: fixes A :: "'b set"   and B :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "A \<subseteq> B"   and "transp_on P B" shows "transp_on P A"
fv\<^sub>s\<^sub>s\<^sub>t_Cons: fixes a :: "('e, 'd) stateful_strand_step"   and A :: "('e, 'd) stateful_strand_step list" shows "fv\<^sub>s\<^sub>s\<^sub>t (a # A) = fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p a \<union> fv\<^sub>s\<^sub>s\<^sub>t A"
dtor1_def: fixes x :: "'b JF1" shows "dtor1 x = F1map id Abs_JF1 Abs_JF2 (str_final1 (Rep_JF1 x))"
map_mono_suffix: fixes xs :: "'c list"   and ys :: "'c list"   and f :: "'c \<Rightarrow> 'd" assumes "suffix xs ys" shows "suffix (map f xs) (map f ys)"
msetext_dersh_imp_mult: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list" assumes "msetext_dersh gt ys xs" shows "(mset xs, mset ys) \<in> mult {(x, y). gt y x}"
simps_17: fixes e :: "'a exp" shows "max_vars (throw e) = max_vars e"
contour_integral_bound_linepath: fixes f :: "complex \<Rightarrow> complex"   and a :: "complex"   and b :: "complex"   and B :: "real" assumes "f contour_integrable_on linepath a b"   and "0 \<le> B"   and "\<And>x. x \<in> closed_segment a b \<Longrightarrow> cmod (f x) \<le> B" shows "cmod (contour_integral (linepath a b) f) \<le> B * cmod (b - a)"
preserves_dom: fixes f :: "'a" assumes "arr f" shows "local.dom (CCC.prod f b) = CCC.prod (local.dom f) b"
Bernstein_coeffs_sum: fixes c :: "real"   and d :: "real"   and P :: "real poly"   and p :: "nat" assumes "c \<noteq> d"   and "degree P \<le> p" shows "P = (\<Sum>j = 0..p. smult (nth_default 0 (Bernstein_coeffs p c d P) j) (Bernstein_Poly j p c d))"
maps_to_n_infinite_seq_res_ring: fixes f :: "nat \<Rightarrow> int"   and k :: "nat" assumes "f \<in> UNIV \<rightarrow> carrier (residue_ring (p ^ k))" shows "\<exists>l. \<forall>m. \<exists>n>m. f n = l"
bounded_linear_left: fixes prod :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and b :: "'e" assumes "bounded_bilinear prod" shows "bounded_linear (\<lambda>a. prod a b)"
symbolic_checker_def: fixes s\<^sub>e :: "real"   and v\<^sub>e :: "real"   and a\<^sub>e :: "real"   and s\<^sub>o :: "real"   and v\<^sub>o :: "real"   and a\<^sub>o :: "real" shows "symbolic_checker s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o \<equiv> let e_stop = - v\<^sub>e / a\<^sub>e; o_stop = - v\<^sub>o / a\<^sub>o in check_precond s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o \<and> \<not> quadroot_in 0 (min e_stop o_stop) (1 / 2 * (a\<^sub>o - a\<^sub>e)) (v\<^sub>o - v\<^sub>e) (s\<^sub>o - s\<^sub>e) \<and> \<not> quadroot_in e_stop o_stop (1 / 2 * a\<^sub>o) v\<^sub>o (s\<^sub>o - movement.p a\<^sub>e v\<^sub>e s\<^sub>e e_stop) \<and> \<not> quadroot_in o_stop e_stop (1 / 2 * a\<^sub>e) v\<^sub>e (s\<^sub>e - movement.p a\<^sub>o v\<^sub>o s\<^sub>o o_stop)"
wfp_on_UNIV: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "wfp_on P UNIV = wfP P"
not_ide_fromArr: fixes Arr :: "'a set"   and x :: "'a" shows "\<not> partial_magma.ide (concrete_category.COMP {False, True} (\<lambda>x y. Collect (graph.path {False, True} Arr (\<lambda>_. False) (\<lambda>_. True) x y)) (\<lambda>_ _ _ g f. f @ g)) (quiver.fromArr Arr x)"
path_trans: fixes weights :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and l1 :: "'a"   and l2 :: "'a"   and xs :: "('a \<times> 'b \<times> 'a) list"   and l3 :: "'a"   and ys :: "('a \<times> 'b \<times> 'a) list" assumes "graph weights"   and "graph.path weights l1 l2 xs"   and "graph.path weights l2 l3 ys" shows "graph.path weights l1 l3 (xs @ ys)"
no_trailing_append: fixes P :: "'f \<Rightarrow> bool"   and xs :: "'f list"   and ys :: "'f list" shows "no_trailing P (xs @ ys) = (no_trailing P ys \<and> (ys = [] \<longrightarrow> no_trailing P xs))"
homeomorphic_trans: fixes S :: "'a set"   and T :: "'b set"   and U :: "'c set" assumes "S homeomorphic T"   and "T homeomorphic U" shows "S homeomorphic U"
arcs_ex_head: fixes i :: "nat"   and j :: "nat"   and xs :: "nat list" shows "\<exists>b. (i, b) \<in> set (arcs i j xs)"
upd'_def: fixes a :: "'a array"   and i :: "integer"   and x :: "'a" shows "Array.upd' a i x = Array.upd (nat_of_integer i) x a \<bind> (\<lambda>_. return ())"