hcju/isabelleps huggingface dataset:
diff_substs_yield_diff_trms: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and x :: "'c"   and \<sigma> :: "('c \<times> 'c trm) list"   and \<eta> :: "('c \<times> 'c trm) list"   and t :: "'c trm" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms"   and "Var x \<lhd> \<sigma> \<noteq> Var x \<lhd> \<eta>"   and "x \<in> vars_of t" shows "t \<lhd> \<sigma> \<noteq> t \<lhd> \<eta>"
basis_extension_def: fixes xs :: "'a vec list" shows "basis_extension xs = (SOME ys. set ys \<subseteq> set (unit_vecs n) \<and> lin_indpt_list (xs @ ys) \<and> length (xs @ ys) = n)"
r_and_prim: shows "prim_recfn 2 r_and"
inner_left_distrib: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner (plus x y) z = inner x z + inner y z"
induced_automorphism_simplex_image_2: fixes C :: "'a set"   and x :: "'a set" assumes "C \<in> g \<turnstile> folding_g.\<C>"   and "order.greater_eq C x" shows "\
 ` x = f ` x"
rtrancl_power: fixes p :: "'a \<times> 'a"   and R :: "('a \<times> 'a) set" shows "(p \<in> R\<^sup>*) = (\<exists>n. p \<in> R ^^ n)"
i_Exec_Stream_Init_prefix: fixes xs :: "'d list"   and input :: "nat \<Rightarrow> 'd"   and trans_fun :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e" assumes "xs \<sqsubseteq> input" shows "f_Exec_Comp_Stream_Init trans_fun xs c \<sqsubseteq> i_Exec_Comp_Stream_Init trans_fun input c"
graph_Entry_transfer: fixes G :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and \<alpha>e1 :: "'f \<Rightarrow> ('h \<times> 'i \<times> 'h) set"   and \<alpha>e2 :: "'g \<Rightarrow> ('h \<times> 'i \<times> 'h) set"   and \<alpha>n1 :: "'f \<Rightarrow> 'h list"   and \<alpha>n2 :: "'g \<Rightarrow> 'h list"   and invar1 :: "'f \<Rightarrow> bool"   and invar2 :: "'g \<Rightarrow> bool"   and inEdges1 :: "'f \<Rightarrow> 'h \<Rightarrow> ('h \<times> 'i) list"   and inEdges2 :: "'g \<Rightarrow> 'h \<Rightarrow> ('h \<times> 'i) list"   and Entry1 :: "'f \<Rightarrow> 'h"   and Entry2 :: "'g \<Rightarrow> 'h" assumes "right_total G"   and "(G ===> (=)) \<alpha>e1 \<alpha>e2"   and "(G ===> (=)) \<alpha>n1 \<alpha>n2"   and "(G ===> (=)) invar1 invar2"   and "(G ===> (=)) inEdges1 inEdges2"   and "(G ===> (=)) Entry1 Entry2" shows "graph_Entry \<alpha>e1 \<alpha>n1 invar1 inEdges1 Entry1 \<longrightarrow> graph_Entry \<alpha>e2 \<alpha>n2 invar2 inEdges2 Entry2"
true_clss_insert: fixes I :: "'b set"   and C :: "'b literal multiset"   and DD :: "'b literal multiset set" shows "(I \<TTurnstile>s insert C DD) = (I \<TTurnstile> C \<and> I \<TTurnstile>s DD)"
num_double: fixes n :: "num" shows "num.Bit0 num.One \<cdot> n = num.Bit0 n"
R: shows "of_int R = (\<Sum>i<n. coeff p i * of_int (a ^ i * b ^ (n - i - 1)))"
Gamma_real_altdef2: fixes x :: "real" shows "Gamma x = Re (Gamma (complex_of_real x))"
simps_81: fixes f :: "'b \<Rightarrow> 'a"   and x1 :: "'b" shows "map_poly f (x1)\<^sub>p = (f x1)\<^sub>p"
kh: fixes x :: "'a \<Rightarrow>\<^sub>0 int" assumes "Poly_Mapping.keys x \<subseteq> S" shows "Poly_Mapping.keys x \<subseteq> S \<and> k (h x) = x"
iT_Plus_cut_ge: fixes I :: "nat set"   and k :: "nat"   and t :: "nat" shows "(I \<oplus> k) \<down>\<ge> t = I \<down>\<ge> (t - k) \<oplus> k"
intros_6: fixes u :: "'c \<Rightarrow> 'b"   and r :: "'c"   and c :: "'c"   and d :: "'b" assumes "u r - u c < d" shows "dbm_entry_val u (Some r) (Some c) (Lt d)"
simps_5: fixes inst :: "inst list"   and funcs :: "cl list"   and tab :: "cl option list list"   and mem :: "mem list"   and globs :: "global list"   and more :: "'a" shows "s.globs \<lparr>inst = inst, funcs = funcs, tab = tab, mem = mem, globs = globs, \<dots> = more\<rparr> = globs"
ctor1_nchotomy: fixes y :: "'d IF1" shows "\<exists>x. y = ctor1 x"
expand_name_propag: fixes n_ns :: "'b node \<times> 'b node set" assumes "((\<forall>nm\<in>incoming (fst n_ns). nm < name (fst n_ns)) \<and> 0 < name (fst n_ns) \<and> (\<forall>q\<in>snd n_ns. name q < name (fst n_ns) \<and> (\<forall>nm\<in>incoming q. nm < name (fst n_ns)))) \<and> expand_name_ident (snd n_ns)" shows "expand n_ns \<le> SPEC (\<lambda>r. (\<forall>q\<in>snd r. name q < fst r \<and> (\<forall>nm'\<in>incoming q. nm' < fst r)) \<and> ((name (fst n_ns) \<le> fst r \<and> name ` snd n_ns \<subseteq> name ` snd r) \<and> name ` snd r = name ` snd n_ns \<union> name ` {nd \<in> snd r. name (fst n_ns) \<le> name nd}) \<and> expand_name_ident (snd r))"
VSigmaE: fixes c :: "V"   and A :: "V"   and B :: "V \<Rightarrow> V" assumes "c \<in> elts (VSigma A B)" obtains x :: "V" and y :: "V" where "x \<in> elts A" and "y \<in> elts (B x)" and "c = \<langle>x, y\<rangle>"
fine_Int: fixes d1 :: "'c \<Rightarrow> 'd set"   and d2 :: "'c \<Rightarrow> 'd set"   and p :: "('c \<times> 'd set) set" shows "((\<lambda>x. d1 x \<inter> d2 x) fine p) = (d1 fine p \<and> d2 fine p)"
derivAssum_def: fixes G :: "((Class \<times> Method \<times> int) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and C :: "Class"   and m :: "Method"   and l :: "int"   and ABI :: "((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)" shows "derivAssum G C m l ABI = (let (A, B, I) = ABI in G \<rhd> \<langle> A \<rangle> C,m,l \<langle> B \<rangle> I)"
simps_16: shows "cospan_shape.comp cospan_shape.TT cospan_shape.BB = cospan_shape.Null"
ordered_resolution_is_complete_ordered_sel: fixes Sel :: "'at set" shows "Complete (ordered_sel_resolvent Sel)"
wp_weaken: fixes x :: "'a"   and p :: "'a"   and q :: "'a" shows "wp x p \<le> wp (x \<cdot> ad q) (d p \<cdot> ad q)"
g2_leq_g1: shows "\<forall>x\<in>cbox a b. g2 x \<le> g1 x"
Eq_llist_2: shows "\<forall>i. enat i < llength u \<longrightarrow> enat i < llength v \<longrightarrow> u ?! i = v ?! i"
greek_embed: fixes r :: "('a \<times> 'a) set"   and a :: "(accent \<times> 'a) list"   and b :: "(accent \<times> 'a) list" assumes "trans r"   and "list_emb (\<lambda>a b. (a, b) \<in> (letter_less r)\<^sup>=) a b" shows "(a, b) \<in> (greek_less r)\<^sup>="
h: shows "convert_poly var p (xs' @ x # xs) = Some (a', b', c')"
Abs_assn_inject: fixes x :: "heap \<times> nat set \<Rightarrow> bool"   and y :: "heap \<times> nat set \<Rightarrow> bool" assumes "x \<in> Collect proper"   and "y \<in> Collect proper" shows "(Abs_assn x = Abs_assn y) = (x = y)"
ext_inject: fixes sa_L :: "'a \<Rightarrow> 'b"   and more :: "'c"   and sa_L' :: "'a \<Rightarrow> 'b"   and more' :: "'c" shows "(\<lparr>sa_L = sa_L, \<dots> = more\<rparr> = \<lparr>sa_L = sa_L', \<dots> = more'\<rparr>) = (sa_L = sa_L' \<and> more = more')"
LBex: fixes f :: "atom fm"   and x :: "real"   and xs :: "real list" assumes "nqfree f"   and "R.I f (x # xs)"   and "\<not> R.I (inf\<^sub>- f) xs"   and "x \<notin> EQ f xs" shows "\<exists>l\<in>{(r - \<langle>cs,xs\<rangle>) / c |r c cs. Less r (c # cs) \<in> set (atoms\<^sub>0 f) \<and> 0 < c}. l < x"
epigraph_mono: fixes x :: "'b"   and y :: "real"   and f :: "'b \<Rightarrow> ereal"   and z :: "real" shows "(x, y) \<in> Epigraph UNIV f \<and> y \<le> z \<longrightarrow> (x, z) \<in> Epigraph UNIV f"
hyps: assumes "\<epsilon> \<notin> set ws" shows "\<^bold>|ws\<^bold>| \<le> \<^bold>|concat ws\<^bold>|"
commute: fixes a :: "'a"   and b :: "'a" shows "inf_class.inf a b = inf_class.inf b a"
FSbImorph_termMOD: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'a"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'b"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b, 'c) model_scheme" shows "FSbImorph h hA termMOD MOD = termFSbImorph h hA MOD"
interp_subseteq_INTERP: fixes C :: "'a literal multiset" shows "interp C \<subseteq> INTERP"
assms_2: shows "i \<in>\<^sub>\<circ> I"
hyps_1: fixes j :: "nat"   and i :: "nat" assumes "x = j - i"   and "i \<le> j" shows "subgraph (S i) (S j)"
phis_measure_update_le': fixes p :: "('node \<times> 'b, 'a) mapping"   and v :: "'b"   and g :: "'g"   and k :: "'node \<times> 'b"   and phis :: "('node \<times> 'b, 'c list) mapping"   and a :: "'a" assumes "phis_measure g v p \<le> phis_measure g v (Mapping.update k [] phis)" shows "phis_measure g v (Mapping.update k a p) \<le> phis_measure g v phis"
xor_commute: fixes conj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and compl :: "'b \<Rightarrow> 'b"   and zero :: "'b"   and one :: "'b"   and a :: "'b"   and b :: "'b" assumes "boolean_algebra conj disj compl zero one" shows "boolean_algebra.xor conj disj compl a b = boolean_algebra.xor conj disj compl b a"
limi_def: fixes T :: "'a set"   and g :: "'b" shows "limi T g \<equiv> {e \<in> T. weight e < g}"
fold_insert_idem: fixes A :: "'a set"   and z :: "'a option"   and x :: "'a" assumes "finite A" shows "Finite_Set.fold MaxR_f z (insert x A) = MaxR_f x (Finite_Set.fold MaxR_f z A)"
sup_left_commute: fixes x :: "'e multiset"   and y :: "'e multiset"   and z :: "'e multiset" shows "x \<union># (y \<union># z) = y \<union># (x \<union># z)"
strict_not_refl_weak: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "rational_preference carrier relation"   and "x \<in> carrier \<and> y \<in> carrier" shows "((y, x) \<notin> relation) = (x \<succeq>[relation] y \<and> (y, x) \<notin> relation)"
pos_divideC_le_eq: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "complex"   and b :: "'b"   and a :: "'b" assumes "class.ordered_complex_vector scaleR scaleC plus zero minus uminus less_eq less"   and "0 < c" shows "less_eq (scaleC (inverse c) b) a = less_eq b (scaleC c a)"
finite_if_finite_subsets_card_bdd: fixes F :: "'c set"   and C :: "nat" assumes "\<And>G. \<lbrakk>G \<subseteq> F; finite G\<rbrakk> \<Longrightarrow> card G \<le> C" shows "finite F \<and> card F \<le> C"
new_vars_private_does_the_thing: fixes \<R> :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set"   and c\<^sub>1\<^sub>A :: "'Com\<^sub>A"   and mds\<^sub>C :: "Mode \<Rightarrow> 'Var\<^sub>C set"   and mem\<^sub>1\<^sub>C :: "'Var\<^sub>C \<Rightarrow> 'Val"   and c\<^sub>1\<^sub>C :: "'Com\<^sub>C"   and c\<^sub>2\<^sub>A :: "'Com\<^sub>A"   and mem\<^sub>2\<^sub>C :: "'Var\<^sub>C \<Rightarrow> 'Val"   and c\<^sub>2\<^sub>C :: "'Com\<^sub>C"   and c\<^sub>1\<^sub>C' :: "'Com\<^sub>C"   and mds\<^sub>C' :: "Mode \<Rightarrow> 'Var\<^sub>C set"   and mem\<^sub>1\<^sub>C' :: "'Var\<^sub>C \<Rightarrow> 'Val"   and c\<^sub>2\<^sub>C' :: "'Com\<^sub>C"   and mem\<^sub>2\<^sub>C' :: "'Var\<^sub>C \<Rightarrow> 'Val" assumes "new_vars_private \<R>"   and "(\<langle>c\<^sub>1\<^sub>A, mds\<^sub>A_of mds\<^sub>C, mem\<^sub>A_of mem\<^sub>1\<^sub>C\<rangle>\<^sub>A, \<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C) \<in> \<R>"   and "(\<langle>c\<^sub>2\<^sub>A, mds\<^sub>A_of mds\<^sub>C, mem\<^sub>A_of mem\<^sub>2\<^sub>C\<rangle>\<^sub>A, \<langle>c\<^sub>2\<^sub>C, mds\<^sub>C, mem\<^sub>2\<^sub>C\<rangle>\<^sub>C) \<in> \<R>"   and "\<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C \<leadsto>\<^sub>C \<langle>c\<^sub>1\<^sub>C', mds\<^sub>C', mem\<^sub>1\<^sub>C'\<rangle>\<^sub>C"   and "\<langle>c\<^sub>2\<^sub>C, mds\<^sub>C, mem\<^sub>2\<^sub>C\<rangle>\<^sub>C \<leadsto>\<^sub>C \<langle>c\<^sub>2\<^sub>C', mds\<^sub>C', mem\<^sub>2\<^sub>C'\<rangle>\<^sub>C"   and "conc.low_mds_eq mds\<^sub>C mem\<^sub>1\<^sub>C mem\<^sub>2\<^sub>C"   and "abs.low_mds_eq (mds\<^sub>A_of mds\<^sub>C') (mem\<^sub>A_of mem\<^sub>1\<^sub>C') (mem\<^sub>A_of mem\<^sub>2\<^sub>C')" shows "conc.low_mds_eq mds\<^sub>C' mem\<^sub>1\<^sub>C' mem\<^sub>2\<^sub>C'"
gcd_eq_gcd_factorial: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Gcd :: "'a set \<Rightarrow> 'a"   and Lcm :: "'a set \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.factorial_semiring_gcd divide plus minus zero times one unit_factor normalize gcd lcm Gcd Lcm" shows "gcd a b = factorial_semiring.gcd_factorial zero times one normalize a b"
take_all: fixes xs :: "'b list"   and n :: "nat" assumes "\<^bold>|xs\<^bold>| \<le> n" shows "take n xs = xs"
supp_Pair: fixes x :: "'e"   and y :: "'f" shows "supp (x, y) = supp x \<union> supp y"
is_sum4sq_int_nat_eq: fixes n :: "nat" shows "is_sum4sq_nat n = is_sum4sq_int (int n)"
lcp_pref': fixes xs :: "'b list"   and ys :: "'b list" shows "xs \<and>\<^sub>p ys \<le>p ys"
reachable_apath: fixes u :: "'a"   and v :: "'a" shows "(u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v) = (\<exists>p. G.apath u p v)"
ind_mult_str_def: fixes S :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e"   and U :: "'d set"   and V :: "'d set" assumes "ind_sheaf S is_open \<FF> \<rho> b add_str mult_str zero_str one_str U" shows "ind_sheaf.ind_mult_str mult_str U V \<equiv> mult_str (U \<inter> V)"
Gauss_Jordan_column_k_iarrays_PA_def: fixes A' :: "'b iarray iarray \<times> nat \<times> 'b iarray iarray"   and k :: "nat" shows "Gauss_Jordan_column_k_iarrays_PA A' k = (let P = fst A'; i = fst (snd A'); A = snd (snd A') in if vector_all_zero_from_index (i, column_iarray k A) \<or> i = nrows_iarray A then (P, i, A) else let Gauss = Gauss_Jordan_in_ij_iarrays_PA (P, A) i k in (fst Gauss, i + 1, snd Gauss))"
fmapWT_unitWT: fixes f :: "'h \<rightarrow> 'e"   and x :: "'h" shows "fmapWT\<cdot>f\<cdot>(unitWT\<cdot>x) = unitWT\<cdot>(f\<cdot>x)"
count_const_lt_components_punit: fixes hs :: "(('a \<Rightarrow>\<^sub>0 'd) \<times> 'e) list" shows "punit.count_const_lt_components hs = (if \<exists>h\<in>set hs. punit.const_lt_component (fst h) = Some () then 1 else 0)"
def_equiv_divisor_set: fixes n :: "nat" shows "divisor_set n = set (divisors_nat n)"
cases: fixes a :: "'a list"   and A :: "'a set" assumes "a \<in> lists A" obtains "aa__ = []"   | a :: "'a" and l :: "'a list" where "aa__ = a # l" and "a \<in> A" and "l \<in> lists A"
prems_4: shows "\<forall>aa. ListMem aa (a # as) \<longrightarrow> agree (fmrestrict_set vs (snd aa)) (fmrestrict_set vs s)"
path_snoc: fixes v :: "'d"   and v' :: "'d"   and pth1 :: "(('d \<times> 'c) \<times> 'd \<times> 'b) list"   and e :: "('d \<times> 'c) \<times> 'd \<times> 'b" shows "path v v' (pth1 @ [e]) = (e \<in> edges \<and> path v (edge_begin e) pth1 \<and> edge_end e = v')"
28269: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- - (x \<sqinter> y) \<squnion> - (- x \<sqinter> z) = - (- x \<sqinter> z)"
card_of_infinite_diff_finite: fixes A :: "'a set"   and B :: "'a set" assumes "infinite A"   and "finite B" shows "|A - B| =o |A|"
ConstFunctor_def: fixes A :: "('a, 'c, 'e) Category_scheme"   and B :: "('b, 'd, 'f) Category_scheme"   and b :: "'b" shows "ConstFunctor A B b \<equiv> MakeFtor (ConstFunctor' A B b)"
lemCProps: fixes m :: "Body"   and v :: "'a" assumes "IOb m"   and "v = c m" shows "(0::'a) < v \<and> (\<forall>x y. (\<exists>p. Ph p \<and> m sees p at x \<and> m sees p at y) = (space2 x y = c m * c m * time2 x y))"
inner_add_left: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "inner (x + y) z = inner x z + inner y z"
distinct_5: fixes lam1 :: "lam"   and lam2 :: "lam"   and name' :: "name"   and lam' :: "lam" shows "lam1 \<degree> lam2 \<noteq> (Lam [name'].lam')"
terminates: shows "\<Gamma>\<turnstile>c \<down> s"
quotient_map_closedin: fixes X :: "'c topology"   and X' :: "'d topology"   and f :: "'c \<Rightarrow> 'd" shows "quotient_map X X' f = (f ` topspace X = topspace X' \<and> (\<forall>U\<subseteq>topspace X'. closedin X {x \<in> topspace X. f x \<in> U} = closedin X' U))"
Irrat_dense_in_real: fixes x :: "real"   and y :: "real" assumes "x < y" shows "\<exists>r\<in>- \<rat>. x < r \<and> r < y"
dghm_HomDom: shows "\<FF>\<lparr>HomDom\<rparr> = \<CC>"
pk: fixes s :: "'a"   and t :: "'a"   and a :: "'b"   and u :: "'a"   and b :: "'b" assumes "(s, t) \<in> L a"   and "(s, u) \<in> R b" shows "(t, u) \<in> conversion'' L R (under r a) O (\<Union> (R ` under q b))\<^sup>= O conversion'' L R (under r a \<union> under r b) O ((\<Union> (L ` under q a))\<inverse>)\<^sup>= O conversion'' L R (under r b)"
assms_2: shows "split_i_j_x i j x = (a, b)"
in_set_remove1: fixes a :: "'b"   and b :: "'b"   and xs :: "'b list" assumes "a \<noteq> b" shows "(a \<in> set (remove1 b xs)) = (a \<in> set xs)"
leg0_composite: shows "fgh.\<mu>\<nu>.leg0 = fgh.\<nu>.leg0 \<cdot> fgh.\<mu>\<nu>.prj\<^sub>0"
intro: fixes r :: "('d \<times> 'd) set"   and s :: "('e \<times> 'e) set" assumes "Well_order r"   and "Well_order s" shows "wo_rel2 r s"
tendsto_dist_iff: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'b"   and F :: "'a filter" shows "(f \<longlongrightarrow> l) F = ((\<lambda>x. dist (f x) l) \<longlongrightarrow> 0) F"
f_def: shows "f \<equiv> \<lambda>y a. if y = x \<and> a \<in> A then x else undefined"
matrix_scaleR: fixes r :: "real" shows "matrix ((*\<^sub>R) r) = mat r"
fmlookup: fixes x :: "('c, 'd) fmap" shows "fmlookup x \<in> {m. finite (dom m)}"
wt_subst_fv_termtype_subterm: fixes x :: "'var"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and y :: "'var" assumes "x \<in> fv (\<theta> y)"   and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>"   and "wf\<^sub>t\<^sub>r\<^sub>m (\<theta> y)" shows "\<Gamma> (Var x) \<sqsubseteq> \<Gamma> (Var y)"
in_hom_def: fixes f :: "'s"   and a :: "'s"   and b :: "'s" shows "\<guillemotleft>f : a \<rightarrow> b\<guillemotright> \<equiv> a \<in> domains f \<and> b \<in> codomains f"
model_to_dimacs_model_all_clauses: fixes n_vars :: "nat"   and f :: "nat formula"   and M :: "nat \<Rightarrow> bool"   and c :: "int list" assumes "1 < n_vars"   and "\<forall>v\<in>atoms f. 0 < v \<and> v < n_vars"   and "is_cnf f"   and "M \<Turnstile> f"   and "c \<in> set (cnf_to_dimacs f)" shows "\<exists>l\<in>set (model_to_dimacs_model M [1..<n_vars]). l \<in> set c"
HBL1_iff: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "bprv (PP \<langle>\<phi>\<rangle>) = prv \<phi>"
tmap_tllist_of_llist: fixes f :: "'g \<Rightarrow> 'e"   and g :: "'h \<Rightarrow> 'f"   and b :: "'h"   and xs :: "'g llist" shows "tmap f g (tllist_of_llist b xs) = tllist_of_llist (g b) (lmap f xs)"
hensels_lemma': fixes f :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int" assumes "f \<in> carrier (UP Zp)"   and "a \<in> carrier Zp"   and "eint 2 * val_Zp (pderiv f \<bullet> a) < val_Zp (f \<bullet> a)" shows "\<exists>!\<alpha>. \<alpha> \<in> carrier Zp \<and> f \<bullet> \<alpha> = \<zero> \<and> val_Zp (pderiv f \<bullet> a) < val_Zp (a \<ominus> \<alpha>)"
conforms_XcptLocD: fixes x :: "abopt"   and s :: "st"   and G :: "prog"   and L :: "lenv"   and a :: "loc" assumes "(x, s)\<Colon>\<preceq>(G, L)"   and "x = Some (Xcpt (Loc a))" shows "G,s\<turnstile>Addr a\<Colon>\<preceq>Class (SXcpt Throwable)"
X_to_Un_supp_def: fixes X :: "'b set" shows "X_to_Un_supp X \<equiv> \<Union> (supp ` X)"
power_mult: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and m :: "nat"   and n :: "nat" assumes "class.monoid_mult one times" shows "power.power one times a (m *n n) = power.power one times (power.power one times a m) n"
tan_sec: fixes x :: "'a" assumes "cos x \<noteq> (0::'a)" shows "(1::'a) + (tan x)\<^sup>2 = (inverse (cos x))\<^sup>2"
meet_left: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y \<sqsubseteq> x"
in_Basis_index_Basis_list: fixes i :: "'b" assumes "i \<in> Basis" shows "i = Basis_list ! index Basis_list i"
strict_trans: fixes a :: "'alt pmf"   and b :: "'alt pmf"   and c :: "'alt pmf" assumes "a \<prec>[Pareto (SD \<circ> R)] b"   and "b \<prec>[Pareto (SD \<circ> R)] c" shows "a \<prec>[Pareto (SD \<circ> R)] c"
select_convs_2: fixes mstate_i :: "nat"   and mstate_m :: "'a mformula"   and mstate_n :: "nat"   and more :: "'b" shows "mstate_m \<lparr>mstate_i = mstate_i, mstate_m = mstate_m, mstate_n = mstate_n, \<dots> = more\<rparr> = mstate_m"
that: fixes Q' :: "('var \<Rightarrow> 'val option) \<Rightarrow> bool" assumes "slice_kind S a = (Q')\<^sub>\<surd>"   and "Q' = (\<lambda>s. False) \<or> Q' = (\<lambda>s. True)" shows "thesis"
prems_1: shows "fmrel R (xm(i \<mapsto>\<^sub>f x)) ym"
deg_le2: fixes n :: "nat" shows "degree (hensel_fpxs2 $ n) \<le> degree h"
Sup_mono: fixes A :: "('c \<times> 'd) set"   and B :: "('c \<times> 'd) set" assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. less_eq_prod a b" shows "less_eq_prod (Sup_prod A) (Sup_prod B)"