hcju/isabelleps huggingface dataset:
hom_mult: fixes x :: "real"   and y :: "real" shows "of_real (x * y) = of_real x * of_real y"
ExcessTable_def: fixes g :: "graph"   and vs :: "nat list" shows "ExcessTable g vs \<equiv> map (\<lambda>v. (v, ExcessAt g v)) (filter (\<lambda>v. 0 < ExcessAt g v) vs)"
singular_face_oriented_simplex: fixes p :: "nat"   and k :: "nat"   and l :: "nat \<Rightarrow> 'c \<Rightarrow> real" assumes "1 \<le> p"   and "k \<le> p" shows "singular_face p k (oriented_simplex p l) = oriented_simplex (p - 1) (\<lambda>j. if j < k then l j else l (Suc j))"
that: assumes "\<not> is_exn \<sigma>"   and "\<sigma>' = \<sigma>"   and "r = Array.length (the_state \<sigma>) a" shows "thesis"
weakPsiCongE_1: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<doteq> Q" shows "\<Psi> \<rhd> P \<approx> Q"
le_imp_pred_less: fixes m :: "nat"   and n :: "nat" assumes "0 < m"   and "m \<le> n" shows "m - Suc 0 < n"
pu_lemma: fixes w1 :: "32 Word.word"   and r1 :: "real"   and w2 :: "32 Word.word"   and r2 :: "real" assumes "w1 \<equiv>\<^sub>U r1"   and "w2 \<equiv>\<^sub>U r2" shows "pu w1 w2 \<equiv>\<^sub>U r1 + r2"
map_eq_Some: shows "[xa # xs [\<mapsto>] y # ys] x = Some Ai"
lappend_LNil_LNil: shows "LNil @@ LNil = LNil"
compatCh_def: fixes ch :: "real + test" shows "Example_PL.compatCh ch \<equiv> \<forall>s t. Example_PL.indisAbbrev s t \<longrightarrow> cval ch s = cval ch t"
constant_rep_designI: fixes \<r> :: "int" assumes "\<And>x. x \<in> verts G \<Longrightarrow> arcs_blocks rep x = \<r>" shows "constant_rep_design (verts G) arcs_blocks \<r>"
valid_init_muaux: fixes L :: "nat set"   and R :: "nat set"   and I :: "\<I>"   and n :: "nat"   and pos :: "bool" assumes "L \<subseteq> R" shows "valid_muaux (init_args I n L R pos) 0 (init_muaux (init_args I n L R pos)) []"
compose_eq: fixes x :: "'c"   and A :: "'c set"   and g :: "'e \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'e" assumes "x \<in> A" shows "compose A g f x = g (f x)"
semiring_norm_51: fixes a :: "'a" shows "a + (0::'a) = a"
LeadsTo_eq_leadsTo: fixes A :: "'c set"   and B :: "'c set" shows "A \<longmapsto>w B = {F. F \<in> reachable F \<inter> A \<longmapsto> reachable F \<inter> B}"
vimage2p_def: fixes f :: "'c \<Rightarrow> 'f"   and g :: "'d \<Rightarrow> 'g"   and R :: "'f \<Rightarrow> 'g \<Rightarrow> 'e" shows "vimage2p f g R = (\<lambda>x y. R (f x) (g y))"
preserves_ide: fixes a :: "'c" assumes "C.ide a" shows "C.ide (src\<^sub>C a)"
sigma_algebra_intersection: fixes \<Omega> :: "'b set"   and A :: "'b set set"   and B :: "'b set set" assumes "sigma_algebra \<Omega> A"   and "sigma_algebra \<Omega> B" shows "sigma_algebra \<Omega> (A \<inter> B)"
descendants_no_head: fixes k :: "'b"   and i :: "'b"   and ps :: "('c, 'b) fm list"   and a :: "'b"   and branch :: "(('c, 'b) fm list \<times> 'b) list"   and xs :: "(nat \<times> nat) set"   and p :: "('c, 'b) fm" assumes "descendants k i ((ps, a) # branch) xs" shows "descendants k i ((p # ps, a) # branch) xs"
GEnd_inner_dirsum_el_decomp_nth: fixes Us :: "'v set list"   and n :: "nat" assumes "\<forall>U\<in>set Us. RModule FG (\<cdot>) U \<and> U \<subseteq> V"   and "add_independentS Us"   and "n < length Us" shows "FGModuleEnd G (\<cdot>) (\<Oplus>U\<leftarrow>Us. U) (\<lambda>a. if a \<in> (\<Oplus>G\<leftarrow>Us. G) then \<Oplus>Us\<leftarrow> a ! n else (0::'v))"
T_get_min_bound: fixes ts :: "'b tree list" assumes "invar ts"   and "ts \<noteq> []" shows "real (T_get_min ts) \<le> log 2 (real (size (mset_heap ts) + 1))"
infinite_Icc: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.dense_linorder less_eq less"   and "less a b" shows "infinite (ord.atLeastAtMost less_eq a b)"
rsquarefree_root_order: fixes p :: "'b poly"   and z :: "'b" assumes "rsquarefree p"   and "poly p z = (0::'b)"   and "p \<noteq> 0" shows "order z p = 1"
finite_Atleast_Atmost_nat: fixes f :: "'d \<Rightarrow> 'c" shows "finite {f x |x. x \<in> UNIV}"
unitary_embed: fixes P :: "'a \<Rightarrow> bool" shows "unitary \<guillemotleft> P \<guillemotright>"
length_inc: fixes y :: "nat list"   and i :: "nat"   and u :: "nat list" shows "length (inc y i u) = length u"
left_neutral: fixes a :: "'a upred" shows "(false \<or> a) = a"
set_cases: fixes e :: "'b"   and a :: "'b environment" assumes "e \<in> set_environment a" obtains z :: "char list \<Rightarrow> 'b option" and x :: "'b option" where "a = Env z" and "x \<in> range z" and "e \<in> set_option x"
stack_consistent_foo_4: fixes as :: "Arity list"   and x :: "Vars.var"   and S :: "stack_elem list" shows "stack_consistent as (Arg x # S) = stack_consistent as S"
ord_eqI_prime_factors: fixes n :: "nat"   and x :: "nat \<times> nat" assumes "\<And>p. p \<in># prime_factorization n \<Longrightarrow> x [^]\<^bsub>S'\<^esub> (n div p) \<noteq> \<one>\<^bsub>S'\<^esub>"   and "x [^]\<^bsub>S'\<^esub> n = \<one>\<^bsub>S'\<^esub>"   and "x \<in> carrier S'"   and "0 < n" shows "S'.ord x = n"
fine_same_iff: fixes \<delta> :: "real \<Rightarrow> real"   and a :: "real"   and D :: "(real \<times> real \<times> real) list" shows "fine \<delta> (a, a) D = (D = [])"
of_nat_simps_1: shows "of_nat 0 = (0::'a)"
ctxt_subst_fresh': fixes X :: "tyvrs"   and \<Gamma> :: "binding list"   and P :: "ty"   and Y :: "tyvrs" assumes "X \<sharp> \<Gamma>"   and "X \<sharp> P" shows "X \<sharp> \<Gamma>[Y \<mapsto> P]\<^sub>e"
mem_single_l_span1: fixes A :: "'b set"   and h :: "'a"   and x :: "'a" assumes "ideal R A"   and "h \<in> carrier M"   and "x \<in> linear_span R M A {h}" shows "\<exists>a\<in>A. x = a \<cdot>\<^sub>s h"
of_int_numeral_le_iff: fixes n :: "num"   and z :: "int" shows "(numeral n \<le> of_int z) = (numeral n \<le> z)"
integral_mono_AE: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "integrable M f"   and "integrable M g"   and "AE x in M. f x \<le> g x" shows "integral\<^sup>L M f \<le> integral\<^sup>L M g"
c_def: shows "c = ''l'' ::= N 0;; ''m'' ::= N 0;; ''r'' ::= Plus (N 1) (V ''x'');; WHILE Less (Plus (N 1) (V ''l'')) (V ''r'') DO (''m'' ::= Div (Plus (V ''l'') (V ''r'')) (N 2);; IF bexp.Not (Less (Times (V ''m'') (V ''m'')) (V ''x'')) THEN ''l'' ::= V ''m'' ELSE ''r'' ::= V ''m'';; ''m'' ::= N 0)"
correctCompositionDiffLevelsA32: shows "correctCompositionDiffLevels sA32"
simps: fixes u :: "'a Vector"   and v :: "'a Vector" shows "u \<oplus> v = \<lparr>tdir = tdir u + tdir v, xdir = xdir u + xdir v, ydir = ydir u + ydir v, zdir = zdir u + zdir v\<rparr>"
set_rewrite2: fixes m :: "nat"   and k :: "nat" shows "{p. p partitions m \<and> sum p {..m} = k \<and> p 1 = 0} = (\<lambda>p i. p (i - 1)) ` {p. p partitions m - k \<and> sum p {..m - k} = k}"
bounded_iff: fixes b :: "'a \<times> 'b"   and c :: "'a \<times> 'b"   and a :: "'a \<times> 'b" shows "less_eq_prod (sup_prod b c) a = (less_eq_prod b a \<and> less_eq_prod c a)"
logN_le_2log_Mn: assumes "m \<noteq> 0"   and "n \<noteq> 0"   and "0 < N" shows "log 2 (real N) \<le> 2 * log 2 (real_of_int (M * int n))"
subst_atm_list_comp_subst: fixes As :: "'a list"   and \<tau> :: "'s"   and \<sigma> :: "'s" shows "As \<cdot>al (\<tau> \<odot> \<sigma>) = As \<cdot>al \<tau> \<cdot>al \<sigma>"
is_processT6: fixes P :: "'a process" shows "\<forall>s X. (s @ [tick], {}) \<in> F P \<longrightarrow> (s, X - {tick}) \<in> F P"
abs_conv_abscissa_weak: shows "abs_conv_abscissa (fds \<chi>) \<le> 1"
filter_plus_ivl_def: fixes i :: "ivl"   and i1 :: "ivl"   and i2 :: "ivl" shows "filter_plus_ivl i i1 i2 = (i1 \<sqinter> minus_ivl i i2, i2 \<sqinter> minus_ivl i i1)"
dom_fun_upd: fixes f :: "'g \<Rightarrow> 'h option"   and x :: "'g"   and y :: "'h option" shows "dom (f(x := y)) = (if y = None then dom f - {x} else insert x (dom f))"
submodule_is_module: fixes N :: "'c set" assumes "LinearCombinations.submodule R N (md N)" shows "module R (N.md N)"
h6: assumes "0 \<le> Av"   and "0 < Bv" shows "(Av + Bv * sqrt Cv \<le> 0) = (Av \<le> 0 \<and> Bv\<^sup>2 * Cv \<le> Av\<^sup>2 \<or> Bv \<le> 0 \<and> Av\<^sup>2 \<le> Bv\<^sup>2 * Cv)"
complete\<^sub>N_I: fixes N :: "'b quasinorm" assumes "\<And>u. \<lbrakk>cauchy_in\<^sub>N N u; \<forall>n. u n \<in> space\<^sub>N N\<rbrakk> \<Longrightarrow> \<exists>x\<in>space\<^sub>N N. tendsto_in\<^sub>N N u x" shows "complete\<^sub>N N"
HInfinite_gt_SReal: fixes x :: "real star"   and y :: "real star" assumes "x \<in> HInfinite"   and "0 < x"   and "y \<in> \<real>" shows "y < x"
isin: fixes t :: "'t"   and x :: "'a" assumes "inv t \<and> sorted (inorder t)" shows "isin t x = (x \<in> list.set (inorder t))"
vec_eq_iff: fixes x :: "'a Matrix.vec"   and y :: "'a Matrix.vec" shows "(x = y) = (dim_vec x = dim_vec y \<and> (\<forall>i<dim_vec y. x $v i = y $v i))"
ffold_finsert2: fixes x :: "'a"   and A :: "'a fset"   and z :: "'b" assumes "x |\<notin>| A" shows "ffold f z (finsert x A) = ffold f (f x z) A"
loopfree_digraph_axioms: shows "loopfree_digraph (app_iso hom G)"
f1: fixes R' :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and gpv :: "('c, 'e, 'f) gpv"   and gpv' :: "('d, 'h, 'g) gpv"   and A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and A' :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "('c, 'e, ('c, 'e, 'f) rpv) generat"   and y :: "('d, 'h, ('d, 'h, 'g) rpv) generat"   and a :: "('c, 'e, 'f) rpv"   and b :: "('d, 'h, 'g) rpv"   and a' :: "'f"   and \<alpha> :: "'g"   and \<beta> :: "'f"   and b' :: "'g" assumes "??.Generative_Probabilistic_Value.foo R' results_gpv results_gpv gpv gpv' A A'"   and "x \<in> set_spmf (the_gpv gpv)"   and "y \<in> set_spmf (the_gpv gpv')"   and "a \<in> generat_conts x"   and "b \<in> generat_conts y"   and "R' a' \<alpha>"   and "R' \<beta> b'" shows "??.Generative_Probabilistic_Value.foo R' results_gpv results_gpv (a a') (b b') A A'"
assms_4: shows "m4 \<in> carrier_mat d2 d2"
pdevs_val_selsort_ccw2: fixes xs :: "(real \<times> real) list"   and e :: "nat \<Rightarrow> real"   and I :: "real set" assumes "distinct xs"   and "e \<in> UNIV \<rightarrow> I" obtains e' :: "nat \<Rightarrow> real" where "e' \<in> UNIV \<rightarrow> I" and "pdevs_val e (pdevs_of_list (ccw.selsort 0 xs)) = pdevs_val e' (pdevs_of_list xs)"
trans: fixes b :: "'a list"   and a :: "'a list"   and c :: "'a list" assumes "prefix b a"   and "prefix c b" shows "prefix c a"
has_binary_products_def: shows "has_binary_products = (\<forall>a0 a1. ide a0 \<and> ide a1 \<longrightarrow> (\<exists>p0 p1. has_as_binary_product a0 a1 p0 p1))"
nonzero_mult_divide_mult_cancel_right2: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.field divide inverse times one plus zero minus uminus"   and "c \<noteq> zero" shows "divide (times a c) (times c b) = divide a b"
sba3_inf_right_dist_sup: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<squnion> y) \<sqinter> z = x \<sqinter> z \<squnion> y \<sqinter> z"
iso_lunit': fixes f :: "'d" assumes "D.ide f" shows "D.iso \<l>\<^sub>D\<^sup>-\<^sup>1[f]"
psubst_new_away'_1: fixes fresh :: "'a"   and t :: "'a FOL_Fitting.term"   and c :: "'a" assumes "new_term fresh t" shows "psubstt (id(fresh := c)) (psubstt (id(c := fresh)) t) = t"
neq_lin: fixes p :: "tm" assumes "tmbound0 p" shows "islin (neq p)"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_val :: "(_) character_data_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_val_locs :: "(_) character_data_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_set_val\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_val set_val_locs"   and "l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr get_child_nodes get_child_nodes_locs" shows "l_set_val_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_val set_val_locs known_ptr get_child_nodes get_child_nodes_locs"
int_pow_pow: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and n :: "int"   and m :: "int" assumes "x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))" shows "(x [^]\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> n) [^]\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> m = x [^]\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> (n * m)"
select_convs_3: fixes ip :: "nat"   and sn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rreqs :: "(nat \<times> nat) set"   and store :: "nat \<Rightarrow> (p \<times> nat list) option"   and msg :: "msg"   and data :: "nat"   and dests :: "nat \<Rightarrow> nat option"   and pre :: "nat set"   and dip :: "nat"   and oip :: "nat"   and hops :: "nat"   and dsn :: "nat"   and dsk :: "k"   and osn :: "nat"   and sip :: "nat"   and more :: "'a" shows "rt \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, pre = pre, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, \<dots> = more\<rparr> = rt"
square_free_rsquarefree: fixes f :: "'b poly" assumes "square_free f" shows "rsquarefree f"
4: shows "\<nexists>parent_ptr. parent_ptr |\<in>| object_ptr_kinds h \<and> node_ptr \<in> set |h \<turnstile> get_child_nodes parent_ptr|\<^sub>r"
crdt_ops_rem_penultimate: fixes xs :: "('a \<times> 'b) list"   and i1 :: "'a"   and r1 :: "'b"   and i2 :: "'a"   and r2 :: "'b"   and deps :: "'b \<Rightarrow> 'a set" assumes "crdt_ops (xs @ [(i1, r1)] @ [(i2, r2)]) deps"   and "\<And>r. r \<in> deps r2 \<Longrightarrow> r \<noteq> i1" shows "crdt_ops (xs @ [(i2, r2)]) deps"
times_bit1_def: fixes x :: "'a bit1"   and y :: "'a bit1" shows "x * y = Abs_bit1' (Rep_bit1 x * Rep_bit1 y)"
midpoint_out: fixes A :: "'p"   and C :: "'p"   and B :: "'p" assumes "A \<noteq> C"   and "B Midpoint A C" shows "A Out B C"
subgroup_Union: fixes I :: "nat \<Rightarrow> 'a set" assumes "\<forall>A\<in>{I n |n. n \<in> UNIV}. subgroup A"   and "\<forall>n. I n \<subseteq> I (n + 1)" shows "subgroup (\<Union> {I n |n. n \<in> UNIV})"
lim_imp_Limsup: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and f0 :: "'b" assumes "F \<noteq> \<bottom>"   and "(f \<longlongrightarrow> f0) F" shows "Limsup F f = f0"
apd1: fixes x :: "('b \<times> 'b) set" shows "x O rel_ar x = {}"
exp_fdiffs: shows "diffs (\<lambda>n. inverse (fact n)) = (\<lambda>n. inverse (fact n))"
conjI_realizer: fixes P :: "'c \<Rightarrow> bool"   and p :: "'c"   and Q :: "'d \<Rightarrow> bool"   and q :: "'d" assumes "P p"   and "Q q" shows "P (fst (p, q)) \<and> Q (snd (p, q))"
simps: fixes \<phi> :: "'b stream \<Rightarrow> bool"   and a :: "'b stream" shows "ev \<phi> a = ((\<exists>xs. a = xs \<and> \<phi> xs) \<or> (\<exists>xs. a = xs \<and> ev \<phi> (stl xs)))"
leg_simps_4: shows "C.dom \<epsilon>.cod.leg1 = \<epsilon>.cod.apex"
rank_Some_card: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and n :: "nat"   and j :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "semi_mojmir_def.rank \<Sigma> \<delta> q\<^sub>0 w x n = Some j" shows "card (semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w x n) = j"
abs_eq: fixes x :: "dflt_size word" shows "uint8_of_uint (Abs_uint x) = Abs_uint8 (ucast x)"
exec_meth_drop: fixes addr2thread_id :: "'a \<Rightarrow> 'b"   and thread_id2addr :: "'b \<Rightarrow> 'a"   and spurious_wakeups :: "bool"   and empty_heap :: "'c"   and allocate :: "'c \<Rightarrow> htype \<Rightarrow> ('c \<times> 'a) set"   and typeof_addr :: "'c \<Rightarrow> 'a \<Rightarrow> htype option"   and heap_read :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> bool"   and heap_write :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> 'c \<Rightarrow> bool"   and ci :: "('a, 'c) check_instr"   and P :: "(nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and ins :: "'a instr list"   and ins' :: "'a instr list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "'b"   and h :: "'c"   and stk :: "'a val list"   and loc :: "'a val list"   and pc :: "nat"   and xcp :: "'a option"   and ta :: "('a, 'b, 'c) jvm_thread_action"   and h' :: "'c"   and stk' :: "'a val list"   and loc' :: "'a val list"   and pc' :: "nat"   and xcp' :: "'a option"   and b :: "nat" assumes "JVM_heap_base.exec_meth addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P (ins @ ins') (shift (length ins) xt) t h (stk, loc, length ins + pc, xcp) ta h' (stk', loc', pc', xcp')"   and "jump_ok ins' 0 b" shows "JVM_heap_base.exec_meth addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P ins' xt t h (stk, loc, pc, xcp) ta h' (stk', loc', pc' - length ins, xcp')"
Span_mem_iff_length_version: fixes R :: "('a, 'b) ring_scheme"   and K :: "'a set"   and Us :: "'a list"   and a :: "'a" assumes "ring R"   and "subfield K R"   and "set Us \<subseteq> carrier R" shows "(a \<in> ring.Span R K Us) = (\<exists>Ks. set Ks \<subseteq> K \<and> length Ks = length Us \<and> a = ring.combine R Ks Us)"
usubst_upd_comm: fixes x :: "'b \<Longrightarrow> 'c"   and y :: "'d \<Longrightarrow> 'c"   and \<sigma> :: "'e \<Rightarrow> 'c"   and u :: "('b, 'e) uexpr"   and v :: "('d, 'e) uexpr" assumes "x \<bowtie> y" shows "\<sigma>(x \<mapsto>\<^sub>s u, y \<mapsto>\<^sub>s v) = \<sigma>(y \<mapsto>\<^sub>s v, x \<mapsto>\<^sub>s u)"
bidirected_digraph_axioms: shows "bidirected_digraph G rev_G"
islimpt_UNIV_iff: fixes x :: "'b" shows "x islimpt UNIV = (\<not> open {x})"
list_slice_less: fixes xs :: "'b list"   and k :: "nat" assumes "length xs < k" shows "list_slice xs k = []"
typ_ok_sig_code: fixes \<Sigma> :: "exesignature"   and ty :: "typ" assumes "exe_osig_conds (exesorts \<Sigma>)" shows "typ_ok_sig (translate_signature \<Sigma>) ty = exetyp_ok_sig \<Sigma> ty"
prove_nonneg: fixes prnt :: "(nat \<times> nat \<times> char list) list"   and i :: "nat"   and p :: "nat"   and fa :: "floatarith"   and XSS :: "(real \<times> real pdevs) list list" assumes "prove_nonneg prnt i p (slp_of_fas [fa]) XSS" shows "\<forall>XS\<in>set XSS. \<forall>xs\<in>Joints XS. 0 \<le> interpret_floatarith fa xs"
widen_C_Object: fixes wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and C :: "String.literal" assumes "wf_prog wf_md P"   and "is_class P C" shows "P \<turnstile> Class C \<le> Class Object"
fun_eq_on_UNIV: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" shows "fun_eq_on f g UNIV = (f = g)"
subseq_rev_drop_many: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "subseq xs ys" shows "subseq xs (ys @ zs)"
min_le_iff_disj: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.linorder less_eq less" shows "less_eq (ord.min less_eq x y) z = (less_eq x z \<or> less_eq y z)"
1: shows "f \<in> mor \<and> m2zf = m2zf"
sep_add_ind_unit: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.heap_sep_algebra plus zero sep_disj"   and "plus x y = zero"   and "sep_disj x y" shows "x = zero"
direct_lim_is_ring: fixes U :: "'a set" assumes "U \<in> I" shows "ring stk2.carrier_direct_lim stk2.add_rel stk2.mult_rel (stk2.class_of U (zero_str2 U)) (stk2.class_of U (one_str2 U))"
lemma_4_2_ii: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a r\<rightarrow> b = c * a r\<rightarrow> c * b"
finite_cyclic_subgroup_order: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a" assumes "Group.group G"   and "x \<in> carrier G" shows "finite (carrier (subgroup_generated G {x})) = (group.ord G x \<noteq> 0)"
Lambda_ext: fixes s :: "ZF"   and f :: "ZF \<Rightarrow> ZF"   and t :: "ZF"   and g :: "ZF \<Rightarrow> ZF" shows "(Lambda s f = Lambda t g) = (s = t \<and> (\<forall>x. x |\<in>| s \<longrightarrow> f x = g x))"
NSDERIV_add: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and Da :: "'a"   and g :: "'a \<Rightarrow> 'a"   and Db :: "'a" assumes "NSDERIV f x :> Da"   and "NSDERIV g x :> Db" shows "NSDERIV (\<lambda>x. f x + g x) x :> Da + Db"
rel_compp: fixes R :: "'b \<Rightarrow> 'e \<Rightarrow> bool"   and S :: "'e \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_H (R OO S) = rel_H R OO rel_H S"
prebicategory_with_units_axioms: shows "prebicategory_with_units (\<cdot>) (\<star>) \<a> some_unit"
homotopy_equivalent_space_contractibility: fixes X :: "'b topology"   and Y :: "'c topology" assumes "X homotopy_equivalent_space Y" shows "contractible_space X = contractible_space Y"
Func_non_emp: fixes B :: "'c set"   and A :: "'d set" assumes "B \<noteq> {}" shows "Func A B \<noteq> {}"
sup_continuous_e2ennreal: fixes f :: "'a \<Rightarrow> ereal" assumes "sup_continuous f" shows "sup_continuous (\<lambda>x. e2ennreal (f x))"
add_less_le_mono: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset"   and d :: "'b multiset" assumes "a \<subset># b"   and "c \<subseteq># d" shows "a + c \<subset># b + d"
a': shows "ide a' \<and> a \<star> a' \<noteq> null \<and> a' \<star> a \<noteq> null \<and> a' \<star> a' \<noteq> null \<and> \<phi> \<star> a' \<noteq> null \<and> Left_a.ide a'"
preserves_isomorphic: fixes f :: "'a"   and g :: "'a" assumes "f \<cong>\<^sub>C g" shows "F f \<cong>\<^sub>D F g"
prems: shows "eq\<rho> (\<A>\<^bsub>a\<^esub> (Terms.Let \<Gamma> e)) \<rho>1 \<rho>2"
abs_inverse: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and equal :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and rep_nat :: "'b \<Rightarrow> nat"   and abs_nat :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "class.nat minus plus zero equal less_eq less rep_nat abs_nat" shows "rep_nat (abs_nat n) = n"
ListRedsVal: fixes e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and v :: "val"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and 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 es :: "expr list"   and es' :: "expr list"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v,s\<^sub>1\<rangle>"   and "P,E \<turnstile> \<langle>es,s\<^sub>1\<rangle> [\<rightarrow>]* \<langle>es',s\<^sub>2\<rangle>" shows "P,E \<turnstile> \<langle>e # es,s\<^sub>0\<rangle> [\<rightarrow>]* \<langle>Val v # es',s\<^sub>2\<rangle>"