hcju/isabelleps huggingface dataset:
Res_def: shows "Res \<equiv> \<lambda>s k. exp (complex_of_real (2 * real k * pi * a) * \<i>) * complex_of_real (2 * real k * pi) powr - s"
properties_def: fixes f :: "'b \<Rightarrow> 'b"   and x0 :: "'b"   and lambda :: "nat"   and mu :: "nat" shows "fx0.properties f x0 lambda mu \<equiv> 0 < lambda \<and> inj_on (fx0.seq' f x0) {0..<mu + lambda} \<and> (\<forall>i\<ge>mu. \<forall>j. fx0.seq' f x0 (i + j * lambda) = fx0.seq' f x0 i)"
intro: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and \<tau>move1 :: "('d, 'b, 'a, 'c, 'e, 'f) \<tau>moves"   and \<tau>move2 :: "('d, 'b, 'g, 'h, 'e, 'f) \<tau>moves"   and bisim :: "'b \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool" assumes "FWdelay_bisimulation_lift_aux final1 r1 final2 r2 \<tau>move1 \<tau>move2"   and "FWdelay_bisimulation_final_base_axioms final1 r1 final2 r2 bisim \<tau>move1 \<tau>move2" shows "FWdelay_bisimulation_final_base final1 r1 final2 r2 bisim \<tau>move1 \<tau>move2"
le_less: fixes y :: "'a"   and x :: "'a" shows "(y \<le> x) = (y < x \<or> x = y)"
Gauss_Jordan_in_ij_PA_def: fixes A' :: "(('a, 'b) vec, 'b) vec \<times> (('a, 'c) vec, 'b) vec"   and i :: "'b"   and j :: "'c" shows "Gauss_Jordan_in_ij_PA A' i j = (let P = fst A'; A = snd A'; n = LEAST n. A $ n $ j \<noteq> (0::'a) \<and> i \<le> n; interchange_A = interchange_rows A i n; interchange_P = interchange_rows P i n; P' = mult_row interchange_P i ((1::'a) / interchange_A $ i $ j) in (\<chi>s. if s = i then P' $ s else row_add P' s i (- interchange_A $ s $ j) $ s, Gauss_Jordan_in_ij A i j))"
1: shows "(SOME t. t \<in> f) \<in> f"
comp_arr_ide: fixes a :: "'b"   and f :: "'b" assumes "C'.ide a"   and "C'.seq f a" shows "C' f a = f"
length_ls_rehash: fixes l :: "('a \<times> 'b) list list" assumes "1 < length l" shows "1 < length (ls_rehash l)"
inject_1: fixes x1 :: "'b \<Rightarrow> ('b, 'c, 'd) sp\<^sub>\<mu>"   and y1 :: "'b \<Rightarrow> ('b, 'c, 'd) sp\<^sub>\<mu>" shows "(Get x1 = Get y1) = (x1 = y1)"
pCons_eq_0_iff: fixes a :: "'b"   and p :: "'b poly" shows "(pCons a p = 0) = (a = (0::'b) \<and> p = 0)"
fragments_basic_com: fixes c' :: "('g, 'h, 'i, 'j) com"   and aft' :: "'j \<Rightarrow> 'h set"   and c :: "('g, 'h, 'i, 'j) com"   and aft :: "'h set" assumes "(c', aft') \<in> fragments c aft" shows "basic_com c'"
obs_elem: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and n :: "'b"   and as :: "'a list"   and n' :: "'b"   and S :: "'b set" assumes "CFG sourcenode targetnode valid_edge Entry"   and "CFG.path sourcenode targetnode valid_edge n as n'"   and "\<forall>nx\<in>set (CFG.sourcenodes sourcenode as). nx \<notin> S"   and "n' \<in> S" shows "n' \<in> CFG.obs sourcenode targetnode valid_edge n S"
finite_Sup_less_iff: fixes X :: "'d set"   and a :: "'d" assumes "finite X"   and "X \<noteq> {}" shows "(Sup X < a) = (\<forall>x\<in>X. x < a)"
brc_invar_def: fixes \<delta> :: "('a, 'b) ta_rule set" shows "brc_invar \<delta> \<equiv> brc_invar_add \<inter> {s. brc_\<alpha> s \<in> br'_invar \<delta>}"
wave: fixes f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "f \<in> Y" shows "wave \<Gamma> f"
simps_4: shows "cospan_shape.comp cospan_shape.BB cospan_shape.BB = cospan_shape.BB"
"3_1": fixes i :: "nat"   and j :: "nat" assumes "\<And>i j k. \<lbrakk>i < length xs; j < length xs; xs ! i = Some k\<rbrakk> \<Longrightarrow> (xs ! j = Some k) = (\<not> tr_lookup T (i + Suc ii) (j + Suc ii))"   and "\<And>a b c. \<lbrakk>a \<le> length T; b \<le> length T; c \<le> length T; \<not> tr_lookup T a b; \<not> tr_lookup T b c\<rbrakk> \<Longrightarrow> \<not> tr_lookup T a c"   and "\<And>x k. \<lbrakk>x \<in> set xs; x = Some k\<rbrakk> \<Longrightarrow> k < length zs"   and "length xs + Suc ii = length T + 1"   and "i < length xs"   and "j < length xs" shows "(fst (mk_eqcl xs zs (Suc ii) T) ! i = fst (mk_eqcl xs zs (Suc ii) T) ! j) = (\<not> tr_lookup T (i + Suc ii) (j + Suc ii))"
assms_3: shows "f \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>"
A1: shows "\<forall>SP SQ. SP \<in>S P \<and> SQ \<in>S Q \<longrightarrow> (\<lbrakk>SP\<rbrakk>, \<lbrakk>SQ\<rbrakk>) \<in> TRel"
fds_abs_convergent_euler_product': fixes f :: "'a fds"   and s :: "'a" assumes "completely_multiplicative_function (fds_nth f)"   and "fds_abs_converges f s" shows "abs_convergent_prod (\<lambda>p. if prime p then inverse ((1::'a) - fds_nth f p / nat_power p s) else (1::'a))"
casts: shows "P \<turnstile> [] Casts [] to vs' "
get_put: fixes x :: "'a \<Longrightarrow> 'b"   and \<sigma> :: "'b" assumes "wb_lens x" shows "put\<^bsub>x\<^esub> \<sigma> (get\<^bsub>x\<^esub> \<sigma>) = \<sigma>"
Un_insert_left: fixes a :: "'b"   and B :: "'b set"   and C :: "'b set" shows "((a::'b) \<triangleright> (B::'b set)) \<union> (C::'b set) = a \<triangleright> B \<union> C"
dom_lookup_keys: fixes al :: "('a, 'b) alist" shows "dom (DAList.lookup al) = AssocList.keys al"
strategyproofness_2: shows "pmf (sds R27) d + pmf (sds R27) c \<le> pmf (sds R19) d + pmf (sds R19) c"
power_diff_sumr2: fixes x :: "'a"   and n :: "nat"   and y :: "'a" shows "x ^ n - y ^ n = (x - y) * (\<Sum>i<n. y ^ (n - Suc i) * x ^ i)"
dom_comp: fixes g :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr"   and f :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "J_C.seq g f" shows "J_C.dom (g \<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] f) = J_C.dom f"
sort_append: fixes xs :: "'a list"   and ys :: "'a list" assumes "\<forall>x\<in>set xs. \<forall>y\<in>set ys. x \<le> y" shows "sort (xs @ ys) = sort xs @ sort ys"
fun_upd_apply: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e"   and y :: "'d"   and z :: "'e" shows "(f(x := y)) z = (if z = x then y else f z)"
Iio_Int_singleton: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and k :: "'a"   and x :: "'a" assumes "class.order less_eq less" shows "ord.lessThan less k \<inter> {x} = (if less x k then {x} else {})"
I_l_closed: fixes I :: "'e set"   and R :: "('e, 'f) ring_scheme"   and a :: "'e"   and x :: "'e" assumes "ideal I R"   and "a \<in> I"   and "x \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> a \<in> I"
interchange_rw: shows "A $ (LEAST n. A $ n $ j \<noteq> (0::'a) \<and> i \<le> n) $ j = interchange_rows A i (LEAST n. A $ n $ j \<noteq> (0::'a) \<and> i \<le> n) $ i $ j"
cases: fixes a1 :: "'f LDCFG_node"   and a2 :: "'g"   and Use :: "'f \<Rightarrow> 'g set"   and E :: "'f"   and X :: "'f"   and H :: "'g set"   and L :: "'g set" assumes "(a1, a2) \<in> lift_Use_set Use E X H L" obtains V :: "'g" and n :: "'f" where "a1 = Node n" and "a2 = V__" and "V__ \<in> Use n"   | V :: "'g" where "a1 = Node E" and "a2 = Va__" and "Va__ \<in> H"   | V :: "'g" where "a1 = Node X" and "a2 = V" and "V \<in> L"
distrib: fixes a :: "'k"   and b :: "'k"   and c :: "'k" shows "(a + b) * c = a * c + b * c"
cobounded2: fixes b :: "'a"   and a :: "'a" shows "b \<le> a + b"
allocationUniverseProperty: fixes x :: "'c"   and X :: "'d set" shows "{(x, X)} - {(x, {})} \<in> allocationsUniverse"
a_get_root_node_locs_def: fixes get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_root_node_locs get_parent_locs = l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_ancestors_locs get_parent_locs"
bsingleton_def: fixes x :: "'e" shows "bsingleton x = binsert x bempty"
add_idem': fixes x :: "'b" shows "x \<squnion> x = x"
oth_class_taut_6_d: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<^bold>\<not>(\<phi> \<^bold>\<or> \<psi>) \<^bold>\<equiv> \<^bold>\<not>\<phi> \<^bold>& \<^bold>\<not>\<psi> in v]"
PiE_arb: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and T :: "'a \<Rightarrow> 'b set"   and x :: "'a" assumes "f \<in> Pi\<^sub>E S T"   and "x \<notin> S" shows "f x = undefined"
well_def_h: shows "\<forall>x. List.member S x \<longrightarrow> x < length subsets"
adapt_stoch_proc_def: fixes F :: "'d \<Rightarrow> 'e measure"   and X :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and N :: "'f measure" shows "adapt_stoch_proc F X N = (\<forall>t. X t \<in> F t \<rightarrow>\<^sub>M N)"
order_eq_one_iff: fixes f :: "'b perm"   and a :: "'b" shows "(order f a = Suc 0) = (a \<notin> affected f)"
finite_nempty_preorder_has_max: fixes B :: "'b set"   and R :: "('b \<times> 'b) set" assumes "finite B"   and "B \<noteq> {}"   and "refl_on B R"   and "trans R"   and "total_on B R" shows "\<exists>x\<in>B. \<forall>y\<in>B. x \<succeq>[R] y"
ex_ex1I: fixes P :: "'b \<Rightarrow> bool" assumes "\<exists>x. P x"   and "\<And>x y. \<lbrakk>P x; P y\<rbrakk> \<Longrightarrow> x = y" shows "\<exists>!x. P x"
cUN_cUn: fixes M :: "'e \<Rightarrow> 'd cset"   and A :: "'e cset"   and B :: "'e cset" shows "cUnion (cimage M (cUn A B)) = cUn (cUnion (cimage M A)) (cUnion (cimage M B))"
isomorphic_implies_hpar_4: fixes f :: "'a"   and f' :: "'a" assumes "f \<cong>\<^sub>C f'" shows "trg\<^sub>C f = trg\<^sub>C f'"
span_in_category_axioms: shows "span_in_category (\<cdot>) (Cod ((\<mu> \<star> \<nu>) \<star> \<pi>))"
inext_nth_card_Max_aux': fixes I :: "nat set" assumes "finite I"   and "I \<noteq> {}" shows "I \<rightarrow> (card I - Suc 0) = Max I"
restrict_preds_to_vars_def: fixes P :: "'BExp set"   and V :: "'Var set" shows "P |` V \<equiv> {e \<in> P. bexp_vars e \<subseteq> V}"
cl_op_nuc: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and cl_op :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.quantale_with_nuc less_eq less cl_op times Inf Sup inf sup bot top" shows "less_eq (times (cl_op x) (cl_op y)) (cl_op (times x y))"
M_Suc_subset: fixes k :: "nat" shows "M (Suc k) \<subseteq> M k"
hta_prod_def: fixes H1 :: "('a, 'c) hashedTa"   and H2 :: "('b, 'c) hashedTa" shows "hta_prod H1 H2 \<equiv> hta_prod' (hta_ensure_idx_s H1) (hta_ensure_idx_sf H2)"
set_sel_2: fixes a :: "'a list"   and x :: "'a" assumes "a \<noteq> []"   and "x \<in> set (tl a)" shows "x \<in> set a"
set_iterator_map_linord_def: fixes it :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<times> 'c \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"   and S0 :: "('b \<times> 'c) set" shows "set_iterator_map_linord it S0 \<equiv> set_iterator_genord it S0 (\<lambda>(k, uu_) (k', uu_). k \<le> k')"
translation_invert: fixes a :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "(+) a ` A = (+) a ` B" shows "A = B"
eq_self_mirror_iff: fixes x :: "'a"   and y :: "'a" shows "(x = mirror_point y x) = (x = y)"
distinct_9: fixes x21 :: "'b tree234"   and x22 :: "'b"   and x23 :: "'b tree234"   and x41 :: "'b tree234"   and x42 :: "'b"   and x43 :: "'b tree234"   and x44 :: "'b"   and x45 :: "'b tree234"   and x46 :: "'b"   and x47 :: "'b tree234" shows "\<langle>x21, x22, x23\<rangle> \<noteq> \<langle>x41, x42, x43, x44, x45, x46, x47\<rangle>"
a: shows "r \<down>l (\<sigma> @ \<tau>) = r \<down>m (r|\<tau>| + r|\<sigma>|)"
flopped_half_chamber_systems_fg: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "ChamberComplex.chamber_system X - f \<turnstile> ChamberComplex.chamber_system X = g \<turnstile> ChamberComplex.chamber_system X"
intros: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry" assumes "[M]\<^bsub>v,n\<^esub> \<subseteq> V"   and "dbm_int M n" shows "valid_dbm M"
hyps: assumes "q \<in> set qs"   and "q \<noteq> 0" shows "q dvd prod_list_var qs"
update_code_def: shows "update_code = RhsEqVal.update rhs_eq_val"
asig_simps_4: fixes asig :: "'h signature" shows "externals asig \<equiv> inputs asig \<union> outputs asig"
match_1: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and h :: "'b"   and g :: "'b"   and f :: "'b" assumes "category C"   and "partial_magma.arr C (C h g)"   and "partial_magma.arr C (C (C h g) f)" shows "partial_magma.arr C (C g f)"
DOWN_UP: fixes x :: "'a" shows "replete_setcat.DOWN (replete_setcat.UP x) = x"
add_frac_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and y :: "'a"   and z :: "'a"   and x :: "'a"   and w :: "'a" assumes "class.field divide inverse times one plus zero minus uminus"   and "y \<noteq> zero"   and "z \<noteq> zero" shows "plus (divide x y) (divide w z) = divide (plus (times x z) (times w y)) (times y z)"
exp_converges: fixes x :: "'a" shows "(\<lambda>n. x ^ n /\<^sub>R fact n) sums exp x"
lens_indepI: fixes x :: "'d \<Longrightarrow> 'f"   and y :: "'e \<Longrightarrow> 'f" assumes "\<And>u v \<sigma>. put\<^bsub>x\<^esub> (put\<^bsub>y\<^esub> \<sigma> v) u = put\<^bsub>y\<^esub> (put\<^bsub>x\<^esub> \<sigma> u) v"   and "\<And>v \<sigma>. get\<^bsub>x\<^esub> (put\<^bsub>y\<^esub> \<sigma> v) = get\<^bsub>x\<^esub> \<sigma>"   and "\<And>u \<sigma>. get\<^bsub>y\<^esub> (put\<^bsub>x\<^esub> \<sigma> u) = get\<^bsub>y\<^esub> \<sigma>" shows "x \<bowtie> y"
iMin_atMost: fixes n :: "nat" shows "iMin {..n} = 0"
distinct_1: fixes x1 :: "Preliminaries.var"   and x21 :: "'a"   and x22 :: "'a trm list" shows "Var x1 \<noteq> Fn x21 x22"
CONSTRAINTI: fixes f :: "'g"   and R :: "('h \<times> 'g) set" shows "CONSTRAINT f R"
simps_7: fixes v :: "char list"   and va :: "aexp"   and vb :: "'a"   and vc :: "'a acom"   and vd :: "'a acom" shows "(v ::= va {vb} \<sqsubseteq> vc;; vd) = False"
3: shows "to_fun g a = to_fun g b \<oplus> deriv g b \<otimes> (a \<ominus> b) \<oplus> to_fun (shift 2 (T\<^bsub>b\<^esub> g)) (a \<ominus> b) \<otimes> (a \<ominus> b) [^] 2"
map_proj_id: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and pol :: "'c \<Rightarrow> 'b \<Rightarrow> epol"   and grdOf :: "('a, 'b) lit list \<Rightarrow> ('a, 'b) lit \<Rightarrow> Preliminaries.var \<Rightarrow> ('a, 'b) lit"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and \<sigma>l :: "'c list"   and al :: "univ list" assumes "ModelIkPolMcalc2C wtFsym wtPsym arOf resOf parOf \<Phi> infTp pol grdOf intT intF intP"   and "list_all2 intT \<sigma>l al" shows "map2 (ModelIkPolMcalc2C.proj intT) \<sigma>l al = al"
rtrancl_power: fixes p :: "'b \<times> 'b"   and R :: "('b \<times> 'b) set" shows "(p \<in> R\<^sup>*) = (\<exists>n. p \<in> R ^^ n)"
m1: fixes m :: "int" assumes "poly_mod_2 m" shows "1 < m"
succs_isNode: fixes x :: "'a" assumes "isNode x" shows "list_all isNode (succs x)"
rel_Grp: fixes A1 :: "'b set"   and f1 :: "'b \<Rightarrow> 'd"   and A2 :: "'c set"   and f2 :: "'c \<Rightarrow> 'e" shows "rel_sum (Grp A1 f1) (Grp A2 f2) = Grp {x. Basic_BNFs.setl x \<subseteq> A1 \<and> Basic_BNFs.setr x \<subseteq> A2} (map_sum f1 f2)"
lessThan_def: fixes less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and u :: "'d" shows "ord.lessThan less u \<equiv> {x. less x u}"
subst_term_preserves_term_ok': fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and x :: "variable"   and \<tau> :: "typ"   and y :: "variable"   and t :: "Core.term" shows "term_ok' \<Sigma> (subst_term [((x, \<tau>), Fv y \<tau>)] t) = term_ok' \<Sigma> t"
integral_iff: fixes R :: "('e, 'f) ring_scheme"   and a :: "'e"   and b :: "'e" assumes "domain R"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "(a \<otimes>\<^bsub>R\<^esub> b = \<zero>\<^bsub>R\<^esub>) = (a = \<zero>\<^bsub>R\<^esub> \<or> b = \<zero>\<^bsub>R\<^esub>)"
mset_rel_def: fixes A :: "('a \<times> 'b) set" shows "\<langle>A\<rangle>mset_rel \<equiv> p2rel (rel_mset (rel2p A))"
doctor_optimal_match_def: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and ds :: "'e set"   and Y :: "'d set" assumes "Contracts Xd Xh Pd Ch" shows "Contracts.doctor_optimal_match Xd Pd Ch ds Y = (Contracts.stable_on Pd Ch ds Y \<and> (\<forall>X. \<forall>x\<in>X. Contracts.stable_on Pd Ch ds X \<longrightarrow> (\<exists>y\<in>Y. (x, y) \<in> Pd (Xd x))))"
nz_2: shows "fds_nth f 1 \<noteq> 0"
map2_empty_iff: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and xs :: "'e list"   and ys :: "'f list" shows "(Map2.map2 f xs ys = []) = (xs = [] \<or> ys = [])"
up_induct: fixes P :: "'b\<^sub>\<bottom> \<Rightarrow> bool"   and x :: "'b\<^sub>\<bottom>" assumes "P \<bottom>"   and "\<And>x. P (up\<cdot>x)" shows "P x"
cont_left_sync: fixes Y :: "nat \<Rightarrow> 'b process"   and A :: "'b set"   and S :: "'b process" assumes "chain Y" shows "((\<Squnion>i. Y i) \<lbrakk>A\<rbrakk> S) = (\<Squnion>i. Y i \<lbrakk>A\<rbrakk> S)"
minus_a_inv: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int" assumes "a \<in> carrier Zp"   and "b \<in> carrier Zp" shows "a \<ominus> b = \<ominus> (b \<ominus> a)"
cases: fixes r :: "gState\<^sub>I" obtains vars :: "(String.literal, variable) assoc_list" and channels :: "channel list" and timeout :: "bool" and procs :: "pState list" and handshake :: "nat" and hsdata :: "integer list" and exclusive :: "nat" and else :: "bool" where "r = \<lparr>gState.vars = vars, channels = channels, timeout = timeout, procs = procs, handshake = handshake, hsdata = hsdata, exclusive = exclusive, else = else\<rparr>"
adm_wo_def: fixes H :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" shows "r.adm_wo H \<equiv> \<forall>f g x. (\<forall>y\<in>r.underS x. f y = g y) \<longrightarrow> H f x = H g x"
in_set_in_list: fixes a :: "('b, 'c) Combinators"   and p :: "('b, 'c) Combinators list"   and l :: "'b set list" assumes "a \<in> set p"   and "all_in_list p l" shows "in_list a l"
indicator_add: fixes A :: "'b set"   and B :: "'b set"   and x :: "'b" assumes "A \<inter> B = {}" shows "indicator A x + indicator B x = indicator (A \<union> B) x"
WTrtThrow: 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 h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and T\<^sub>r :: "ty"   and T :: "ty" assumes "P,E,h \<turnstile> e : T\<^sub>r"   and "is_refT T\<^sub>r" shows "P,E,h \<turnstile> throw e : T"
approx_reorient: fixes x :: "'a star"   and y :: "'a star" shows "(x \<approx> y) = (y \<approx> x)"
dependent_single: fixes x :: "('a, 'b) vec" shows "vec.dependent {x} = (x = 0)"
interfree_def: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and \<Theta> :: "'a \<Rightarrow> ('b, 'a, 'c) ann list option"   and F :: "'c set"   and Ps :: "(('b, 'a, 'c) ann \<times> 'b set \<times> 'b set) list"   and Ts :: "('b, 'a, 'c) com list" shows "interfree \<Gamma> \<Theta> F Ps Ts \<equiv> \<forall>i j. i < length Ts \<and> j < length Ts \<and> i \<noteq> j \<longrightarrow> interfree_aux \<Gamma> \<Theta> F (Ts ! i, Ps ! i, Ts ! j, fst (Ps ! j))"
return_reads: fixes x :: "'c"   and h :: "'a"   and h' :: "'a" shows "reads {} (return x) h h'"
Image_subset_eq: fixes r :: "('c \<times> 'b) set"   and A :: "'c set"   and B :: "'b set" shows "(r `` A \<subseteq> B) = (A \<subseteq> - r\<inverse> `` (- B))"
G: shows "graph_of_mat a = map_prod Bij_Nat.to_nat Bij_Nat.to_nat ` G"
assms_1: shows "\<psi> \<in> \<R>\<^sup>2"
strict_reflclp_conv: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "strict P\<^sup>=\<^sup>= = strict P"
in_pdata_pairs_to_listI2: fixes f :: "('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e"   and g :: "('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e"   and ps :: "((('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) \<times> ('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list" assumes "(f, g) \<in> set ps" shows "monom_mult ((1::'d) / lc (fst g)) (lcs (lp (fst f)) (lp (fst g)) - lp (fst g)) (fst g) \<in> set (pdata_pairs_to_list ps)"
cexpr_sem_cexpr_subst_val: fixes \<sigma> :: "nat \<Rightarrow> val"   and e :: "cexpr"   and v :: "val" shows "cexpr_sem \<sigma> (cexpr_subst_val e v) = cexpr_sem (case_nat v \<sigma>) e"
induct: fixes P :: "'a msllist \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "'a msllist"   and a1 :: "nat" assumes "\<And>n. P MSLNil n"   and "\<And>x xs. P (MSLCons x xs) 0"   and "\<And>x xs n. P xs n \<Longrightarrow> P (MSLCons x xs) (Suc n)" shows "P a0 a1"
card_Mi_le_floor_div_2_Vi: fixes L :: "'b \<Rightarrow> nat"   and E :: "('b \<times> 'b) set"   and V :: "'b set"   and M :: "('b \<times> 'b) set"   and i :: "nat" assumes "OSC L E \<and> matching V E M \<and> 1 < i" shows "card (matching_i i V E M L) \<le> card (V_i i V E M L) div 2"
infer_fnE: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and x :: "'a"   and T :: "PreSimplyTyped.type"   and A :: "'a trm"   and \<tau> :: "PreSimplyTyped.type" assumes "infer \<Gamma> (Fn x T A) = Some \<tau>" shows "\<exists>\<sigma>. \<tau> = TArr T \<sigma> \<and> infer (\<Gamma>(x \<mapsto> T)) A = Some \<sigma>"
x_bigger_min: shows "Min_mset (arr_mset arr l\<^sub>0 m) \<le> x"
assoc_comp: fixes U :: "'a set set"   and V :: "'a set set"   and W :: "'a set set"   and x :: "'a set \<Rightarrow> ('a \<times> 'a) set" assumes "is_zariski_open U"   and "is_zariski_open V"   and "is_zariski_open W"   and "V \<subseteq> U"   and "W \<subseteq> V"   and "x \<in> local.im_sheaf U" shows "im_sheaf_morphisms U W x = (im_sheaf_morphisms V W \<circ> im_sheaf_morphisms U V) x"
split_minD: fixes t :: "'c tree234"   and x :: "'c"   and t' :: "'c up\<^sub>d" assumes "split_min t = (x, t')"   and "bal t"   and "0 < height t" shows "x # inorder (tree\<^sub>d t') = inorder t"
measurable_case_nat_Suc_PiM: fixes A :: "nat set"   and M :: "'a measure"   and N :: "nat \<Rightarrow> 'a measure" shows "(\<lambda>\<sigma>. \<sigma> \<circ> Suc) \<in> Pi\<^sub>M (Suc ` A) (case_nat M N) \<rightarrow>\<^sub>M Pi\<^sub>M A N"
the_wall_betw_self_empty: fixes C :: "'a set" shows "THE_default {} (\<lambda>H. H \<in> (\<Union>(f, g)\<in>foldpairs. {{f \<turnstile> \<C>, g \<turnstile> \<C>}}) \<and> separated_by H C C) = {}"
card_distinct: fixes xs :: "'d list" assumes "card (set xs) = length xs" shows "distinct xs"
not_summable_infsetsum_eq: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "\<not> f abs_summable_on A" shows "infsetsum f A = (0::'c)"
target: fixes T1 :: "'procT"   and T2 :: "'procT"   and TRel :: "('procT \<times> 'procT) set" assumes "(T1, T2) \<in> TRel" shows "TargetTerm T1 \<R>\<lbrakk>\<cdot>\<rbrakk>LT<TRel> TargetTerm T2"
if_SE_D2': fixes P :: "'d \<Rightarrow> bool"   and \<sigma> :: "'d"   and B\<^sub>1 :: "'d \<Rightarrow> ('e \<times> 'd) option"   and B\<^sub>2 :: "'d \<Rightarrow> ('e \<times> 'd) option"   and M :: "'d \<Rightarrow> (bool \<times> 'd) option" assumes "\<not> P \<sigma>" shows "(\<sigma> \<Turnstile> _if_SE P B\<^sub>1 B\<^sub>2 ;- M) = (\<sigma> \<Turnstile> B\<^sub>2 ;- M)"