hcju/isabelleps huggingface dataset:
l7_22_aux_R2: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A2 :: "'a"   and C :: "'a"   and A1 :: "'a"   and B1 :: "'a"   and B2 :: "'a"   and M1 :: "'a"   and M2 :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A2 \<noteq> C"   and "Bet A1 C A2"   and "Bet B1 C B2"   and "Cong C A1 C B1"   and "Cong C A2 C B2"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong M1 A1 B1"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong M2 A2 B2"   and "Tarski_neutral_dimensionless.Le Bet Cong C A1 C A2" shows "Bet M1 C M2"
pentagon: fixes a :: "'d"   and b :: "'d"   and c :: "'d"   and d :: "'d" assumes "D.ide a"   and "D.ide b"   and "D.ide c"   and "D.ide d" shows "(a \<otimes>\<^sub>D \<a>\<^sub>D[b, c, d]) \<cdot>\<^sub>D \<a>\<^sub>D[a, b \<otimes>\<^sub>D c, d] \<cdot>\<^sub>D (\<a>\<^sub>D[a, b, c] \<otimes>\<^sub>D d) = \<a>\<^sub>D[a, b, c \<otimes>\<^sub>D d] \<cdot>\<^sub>D \<a>\<^sub>D[a \<otimes>\<^sub>D b, c, d]"
leq_linear: fixes leq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.linear_order leq" shows "leq x y \<or> leq y x"
assign_r_comp: fixes x :: "'c \<Longrightarrow> 'a"   and u :: "('c, 'a) uexpr"   and P :: "('a, 'b) urel" shows "x := u ;; P = P\<lbrakk>\<lceil>u\<rceil>\<^sub></$x\<rbrakk>"
mult_on_union: fixes M :: "'a multiset"   and N :: "'a multiset"   and r :: "('a \<times> 'a) set"   and K :: "'a multiset" assumes "(M, N) \<in> mult r" shows "(K + M, K + N) \<in> mult r"
valid_path_join_D1: fixes g1 :: "real \<Rightarrow> 'b"   and g2 :: "real \<Rightarrow> 'b" assumes "valid_path (g1 +++ g2)" shows "valid_path g1"
list_to_mat_to_cols_list: fixes nr :: "nat"   and nc :: "nat"   and l :: "complex list list" assumes "Matrix_Legacy.mat nr nc l" shows "mat_to_cols_list (Tensor.mat_of_cols_list nr l) = l"
img_point_elem_set: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and x :: "'b"   and a :: "'b" assumes "set_category S \<AA>"   and "partial_magma.in_hom S x (set_category_data.unity S) a" shows "set_category.img S \<AA> x \<in> set_category_data.set S (set_category.some_img S \<AA>) a"
simps_6: fixes x1 :: "'c" shows "\<infinity> \<noteq> Le x1"
imp_imp_compat_eqvL: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<chi> \<in> fmla" shows "prv (imp (eqv \<phi>1 \<phi>2) (eqv (imp \<phi>1 \<chi>) (imp \<phi>2 \<chi>)))"
prems: fixes i :: "nat"   and j :: "nat" assumes "i < j"   and "i < length (x # xs)"   and "j < length (x # xs)" shows "(x # xs) ! i \<noteq> (x # xs) ! j"
ttree_restr_is_empty: fixes t :: "'b ttree"   and S :: "'b set" assumes "carrier t \<inter> S = {}" shows "ttree_restr S t = TTree.empty"
ts__coplanar: fixes A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "A B TS C D" shows "Coplanar A B C D"
pag: shows "path_image \<gamma> \<subseteq> {w. b < a \<bullet> w}"
in_code_status_status_rel_iff_5: fixes a :: "'a code_status" shows "((a, FAILED) \<in> code_status_status_rel) = is_cfailed a"
annotate_code_def: fixes P :: "'b J_prog"   and E :: "String.literal \<Rightarrow> ty option"   and e :: "'b expr" shows "annotate_code P E e = THE_default e (Anno_code P E e)"
subcategory_VVV: shows "subcategory B.VxVxV.comp (\<lambda>\<tau>\<mu>\<nu>. B.arr (fst \<tau>\<mu>\<nu>) \<and> B.VV.arr (snd \<tau>\<mu>\<nu>) \<and> src\<^sub>B (fst \<tau>\<mu>\<nu>) = trg\<^sub>B (fst (snd \<tau>\<mu>\<nu>)))"
rep_eq: fixes x :: "'a"   and xa :: "'b poly" shows "lookup (pm_of_poly x xa) = (\<lambda>t. poly.coeff xa (lookup t x) when t \<in> .[{x}])"
strategy_attracts_def: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and A :: "'a set"   and W :: "'a set" shows "G'.strategy_attracts p \<sigma> A W \<equiv> \<forall>v0\<in>A. G'.strategy_attracts_via p \<sigma> v0 A W"
fresh_aux: fixes c :: "name"   and a :: "name"   and x :: "'a"   and b :: "name" assumes "c \<noteq> a"   and "a \<sharp> x"   and "c \<sharp> x" shows "c \<sharp> [(a, b)] \<bullet> x"
lemDirectionReverse: fixes q :: "'a Point"   and p :: "'a Point" shows "from q to p = ~ from p to q"
fromMOD_basic_simps_1: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model"   and s :: "'sort"   and X'X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<times> 'c" shows "igWls (fromMOD MOD) s X'X = (wls s (fst X'X) \<and> gWls MOD s (snd X'X))"
nwprops: shows "qty_mult_comp pf1 qty x n w \<noteq> 0"
inject_4: fixes x3 :: "trm"   and x1 :: "name"   and x2 :: "trm"   and y3 :: "trm"   and y1 :: "name"   and y2 :: "trm" shows "(x3 to x1 in x2 = y3 to y1 in y2) = (x3 = y3 \<and> [x1].x2 = [y1].y2)"
nres_relD: fixes c :: "'g nres"   and a :: "'h nres"   and R :: "('g \<times> 'h) set" assumes "(c, a) \<in> \<langle>R\<rangle>nres_rel" shows "c \<le> \<Down> R a"
assms_3: shows "lex_ord_raw ko f xs ys = Some Eq"
fixp_induct: fixes P :: "'a \<times> 'b \<Rightarrow> bool"   and f :: "'a \<times> 'b \<Rightarrow> 'a \<times> 'b" assumes "ccpo.admissible (prod_lub luba lubb) (rel_prod orda ordb) P"   and "monotone (rel_prod orda ordb) (rel_prod orda ordb) f"   and "P (prod_lub luba lubb {})"   and "\<And>x. P x \<Longrightarrow> P (f x)" shows "P (ab.fixp f)"
le_less_trans: fixes x :: "'b word"   and y :: "'b word"   and z :: "'b word" assumes "x \<le>s y"   and "y <s z" shows "x <s z"
perm_5: fixes pi :: "(name \<times> name) list"   and x2 :: "('d, 'e, 'f) psi"   and x1 :: "('d, 'e, 'f) psi" shows "pi \<bullet> x2 \<parallel> x1 = (pi \<bullet> x2) \<parallel> (pi \<bullet> x1)"
tSplice_spr_jview_not_a: fixes t :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and t' :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and a :: "'a"   and a' :: "'a" assumes "tObsC t = tObsC t'"   and "a \<noteq> a'" shows "spr_jview a' (t \<^bsub>\<^esub>\<bowtie>\<^bsub>a\<^esub> t') = spr_jview a' t"
strict_trans: fixes b :: "'c"   and a :: "'c"   and c :: "'c" assumes "b < a"   and "c < b" shows "c < a"
hyps_2: shows "degree (p \<circ>\<^sub>p q) = degree p * degree q"
prems_5: shows "P \<turnstile> Ts' [\<le>] Ts"
elements_with_ord_restrict_totatives: fixes d :: "nat" assumes "0 < d" shows "{x \<in> {..<n}. ord n x = d} = {x \<in> totatives n. ord n x = d}"
closest_supchamberD_2: fixes F :: "'a set"   and D :: "'a set" assumes "F \<in> X"   and "chamber D" shows "order.greater_eq (closest_supchamber F D) F"
i_4: fixes a :: "'c" assumes "a \<in> insert x F" shows "f a \<otimes> g a = h a"
hpowr_Suc: fixes r :: "real star"   and n :: "nat" shows "r ^ Suc n = r * r ^ n"
k': shows "k < length (snd (dioph_dfs n ks l)) \<and> snd (dioph_dfs n ks l) ! k = m"
typeI_cube_line_integral_exists_boundary: shows "\<forall>(k, \<gamma>)\<in>two_chain_boundary two_chain. line_integral_exists F {i} \<gamma>"
prems_2: shows "DIM('a) + 1 \<le> Suc n"
ind_mult_str_def: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and U :: "'c set"   and V :: "'c 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)"
bi_unique_intersect_l: fixes T :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'e set"   and a' :: "'f set"   and b :: "'e set"   and b' :: "'f set"   and xl :: "'e set" assumes "bi_unique T"   and "rel_set T a a'"   and "rel_set T b b'"   and "rel_set T xl (a' \<inter> b')" shows "a \<inter> b = xl"
IH: fixes l2 :: "'a list" assumes "sorted l1"   and "sorted l2" shows "sorted (merge l1 l2) \<and> mset (merge l1 l2) = mset l1 + mset l2 \<and> set (merge l1 l2) = set l1 \<union> set l2"
strictly_subsumes_def: fixes subst_atm :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and C :: "'a literal multiset"   and D :: "'a literal multiset" shows "substitution_ops.strictly_subsumes subst_atm C D = strict (substitution_ops.subsumes subst_atm) C D"
rtos_r_s: fixes r :: "nat"   and s :: "nat" assumes "0 < r"   and "0 < s" shows "rtos r s (r * s - Suc 0) = r * s - Suc 0"
thetaIfZOT_ZOretrT: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a" assumes "PL_Indis indis" shows "PL_Indis.thetaIfZOT tval aval indis \<subseteq> PL_Indis.ZOretrT tval aval indis (PL_Indis.thetaIfZOT tval aval indis \<union> PL_Indis.ZObisT tval aval indis)"
UN_subset_iff: fixes A :: "'c \<Rightarrow> 'b set"   and I :: "'c set"   and B :: "'b set" shows "(\<Union> (A ` I) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
supports_finite: fixes S :: "atom set"   and x :: "'a" assumes "S supports x"   and "finite S" shows "finite (supp x)"
unit_circle_def: shows "unit_circle \<equiv> abs_circline unit_circle_clmat"
param_map: fixes R1 :: "('a \<times> 'c) set"   and R2 :: "('b \<times> 'd) set" shows "(map, map) \<in> (R1 \<rightarrow> R2) \<rightarrow> \<langle>R1\<rangle>list_rel \<rightarrow> \<langle>R2\<rangle>list_rel"
gfeq: shows "\<forall>x\<in>h ` {0..1}. g (f x) = x"
distinct_cycle_edges: fixes G :: "('a, 'b) graph_scheme"   and p :: "'a list" assumes "cycle G p" shows "distinct (cycle_edges p)"
R2: fixes p :: "('a \<times> 'a) set"   and x :: "('a \<times> 'a) set"   and q :: "('a \<times> 'a) set" assumes "rel_kat.H p x q" shows "x \<subseteq> rel_R p q"
H2_3: shows "AE x in M. 0 \<le> g2 x \<and> g2 x \<le> max 0 (f1 x + C x)"
assms_5: shows "\<forall>i. \<not> is_zero_row i A \<longrightarrow> (\<forall>j<i. A $ j $ (LEAST n. A $ i $ n \<noteq> (0::'a)) \<in> residues (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a))))"
assms: shows "P \<Longrightarrow>\<^sub>l\<^sup>^\<alpha> \<prec> P'"
var_dist_cong: fixes s :: "dB"   and t :: "dB"   and vs :: "nat list" assumes "s \<leftrightarrow> t" shows "var_dist vs s \<leftrightarrow> var_dist vs t"
map_snd_alist_of: fixes q :: "('d, 'c) pq" shows "map (the \<circ> priority q) |q| = map snd (alist_of q)"
Eq_cong: fixes H :: "fm set"   and t :: "tm"   and t' :: "tm"   and u :: "tm"   and u' :: "tm" assumes "H \<turnstile> t EQ t'"   and "H \<turnstile> u EQ u'" shows "H \<turnstile> t EQ u IFF t' EQ u'"
vsv_vimageI2': fixes b :: "V"   and a :: "V" assumes "b = \<FF>\<lparr>ArrMap\<rparr>\<lparr>a\<rparr>"   and "a \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>" shows "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (\<FF>\<lparr>ArrMap\<rparr>)"
Dags_eq_hp_eq: fixes p :: "ref"   and l :: "ref \<Rightarrow> ref"   and r :: "ref \<Rightarrow> ref"   and t :: "dag"   and p' :: "ref"   and l' :: "ref \<Rightarrow> ref"   and r' :: "ref \<Rightarrow> ref" assumes "Dag p l r t"   and "Dag p' l' r' t" shows "p' = p \<and> (\<forall>no\<in>set_of t. l' no = l no \<and> r' no = r no)"
fresh_in_nat_def: fixes S :: "nat set" shows "fresh_in S \<equiv> if Set.is_empty S then 0 else Max S + 1"
isomorphic_def: fixes a :: "'d"   and a' :: "'d" shows "D.isomorphic a a' = (\<exists>f. \<guillemotleft>f : a \<Rightarrow>\<^sub>D a'\<guillemotright> \<and> D.iso f)"
Image_UN: fixes r :: "('i \<times> 'h) set"   and B :: "'j \<Rightarrow> 'i set"   and A :: "'j set" shows "r `` \<Union> (B ` A) = (\<Union>x\<in>A. r `` B x)"
prems_4: shows "Marker \<in> set (msgs (S t i) cid)"
carrier_Fheap': fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and a :: "Arity" shows "carrier (Theap \<Gamma> e\<cdot>a) = edom (Aheap \<Gamma> e\<cdot>a)"
length_rev: fixes xs :: "'a list" shows "length (rev xs) = length xs"
strict_sorted_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and l :: "'a list" assumes "class.linorder less_eq less" shows "linorder.strict_sorted less l = (linorder.sorted less_eq l \<and> distinct l)"
arcs_ends_conv: fixes G :: "('c, 'd) pre_digraph" shows "arcs_ends G = (\<lambda>e. (tail G e, head G e)) ` arcs G"
lderivs_Plus: fixes xs :: "'a list"   and r :: "'b rexp"   and s :: "'b rexp" shows "lderivs xs (Plus r s) = Plus (lderivs xs r) (lderivs xs s)"
valid_join_mmsaux_unfolded: fixes args :: "Monitor.args"   and cur :: "nat"   and nt :: "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 X :: "'b option list set" assumes "valid_mmsaux args cur (nt, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since) auxlist"   and "table (args_n args) (args_L args) X" shows "valid_mmsaux args cur (join_mmsaux args X (nt, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since)) (map (\<lambda>(t, rel). (t, join rel (args_pos args) X)) auxlist)"
simps_3: fixes m :: "'b match_expr" shows "has_primitive (MatchNot m) = has_primitive m"
Max_wrt_among_subset_1: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "Order_Predicates.preorder_on carrier le" shows "Max_wrt_among le A \<subseteq> carrier"
finite_varsC: fixes c :: "('c, 'd) lit list" shows "finite (varsC c)"
simps_3: fixes var :: "nat"   and p :: "real mpoly" shows "substNegInfinity var (Leq p) = Or (alternateNegInfinity p var) (allZero p var)"
Hoare_def: fixes p :: "'a"   and x :: "'a"   and q :: "'a" shows "H p x q = \<lparr>\<tt>\<tt> p\<rparr>x\<lparr>x \<cdot> \<tt>\<tt> q\<rparr>"
fst_lem1: fixes a :: "'a set"   and c :: "'a set"   and b :: "'a set"   and d :: "'a set" assumes "(a, c) = (b, d)" shows "a = b"
dims2_def: shows "st.dims2 = nths dims {Suc k..<n}"
sees_methods_Object: fixes P :: "'a prog"   and D :: "String.literal"   and fs :: "fdecl list"   and ms :: "'a option mdecl list"   and Mm :: "String.literal \<Rightarrow> ((ty list \<times> ty \<times> 'a option) \<times> String.literal) option" assumes "class P Object = \<lfloor>(D, fs, ms)\<rfloor>"   and "Mm = map_option (\<lambda>m. (m, Object)) \<circ> map_of ms" shows "P \<turnstile> Object sees_methods Mm"
chop_max: fixes i :: "nat_int"   and j :: "nat_int"   and k :: "nat_int" shows "N_Chop(i,j,k) \<and> consec j k \<longrightarrow> maximum i = maximum k"
adt_wgraph_axioms: shows "adt_wgraph G.\<alpha> G.invar G_succ G_empty G_add_edge"
XG_def: fixes es :: "nat set set" shows "XG es = {c \<in> ucycles (edge_ugraph es). uwalk_length c = k}"
f: fixes m :: "'a measure" assumes "m \<in> M" shows "f \<in> N \<rightarrow>\<^sub>M m"
cancel1_same: fixes xs :: "'d list" assumes "xs \<noteq> []" shows "cancel1 (xs @ xs) xs"
qq3: shows "supp r \<subseteq> q \<bullet> as \<union> bs"
uminus_closed: fixes a :: "'vs" assumes "a \<in> U\<^sub>V\<^sub>S" shows "-\<^sub>V\<^sub>S a \<in> U\<^sub>V\<^sub>S"
reduce_clauses_in_P: fixes C :: "'a literal list"   and N :: "('a literal list \<times> nat) list"   and P :: "('a literal list \<times> nat) list"   and P' :: "('a literal list \<times> nat) list"   and Q :: "('a literal list \<times> nat) list"   and n :: "nat" assumes "C \<in> fst ` set N"   and "\<forall>(E, k)\<in>set P. is_irreducible [C] E" shows "wstate_of_dstate (N, P @ P', Q, n) \<leadsto>\<^sub>w\<^sup>* wstate_of_dstate (N, P @ reduce_all C P', Q, n)"
sints_def: fixes n :: "nat" shows "sints n = range (signed_take_bit (n - 1))"
is_parserI: fixes p :: "char list \<Rightarrow> char list + 'a \<times> char list" assumes "\<And>s r x. p s = Inr (x, r) \<Longrightarrow> length r \<le> length s" shows "is_parser p"
stri_mul_ext_map: fixes ss :: "'a list"   and ts :: "'a list"   and order :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and order' :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>s t. \<lbrakk>s \<in> set ss; t \<in> set ts; fst (order s t)\<rbrakk> \<Longrightarrow> fst (order' (f s) (f t))"   and "\<And>s t. \<lbrakk>s \<in> set ss; t \<in> set ts; snd (order s t)\<rbrakk> \<Longrightarrow> snd (order' (f s) (f t))"   and "fst (mul_ext order ss ts)" shows "fst (mul_ext order' (map f ss) (map f ts))"
iso_points_ss_card: fixes ps :: "'a set" assumes "ps \<subseteq> \<V>" shows "card ps = card (\<pi> ` ps)"
uminus_assertion: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "x \<in> mbt_algebra.assertion one times inf less_eq dual top" shows "mbt_algebra.neg_assert one times inf dual bot x \<in> mbt_algebra.assertion one times inf less_eq dual top"
lemma_3_2: fixes i :: "int"   and j :: "int" assumes "i \<le> j" shows "K i (j + 1) \<le> K (i + 1) (j + 1)"
simps: fixes a :: "'c"   and g :: "'c \<Rightarrow> ('d, 'c) step" shows "(a \<in> terminates_on g) = ((\<exists>s. a = s \<and> g s = Done) \<or> (\<exists>s s'. a = s \<and> g s = Skip s' \<and> s' \<in> terminates_on g) \<or> (\<exists>s aa s'. a = s \<and> g s = Yield aa s' \<and> s' \<in> terminates_on g))"
strict_trans: fixes b :: "('a, 'b) Term.term"   and a :: "('a, 'b) Term.term"   and c :: "('a, 'b) Term.term" assumes "b \<sqsubset> a"   and "c \<sqsubset> b" shows "c \<sqsubset> a"
is_process_spec: fixes P :: "('a event list \<times> 'a event set) set \<times> 'a event list set" shows "is_process P = (([], {}) \<in> FAILURES P \<and> (\<forall>s X. (s, X) \<in> FAILURES P \<longrightarrow> front_tickFree s) \<and> (\<forall>s t. (s @ t, {}) \<notin> FAILURES P \<or> (s, {}) \<in> FAILURES P) \<and> (\<forall>s X Y. (s, Y) \<notin> FAILURES P \<or> \<not> X \<subseteq> Y \<or> (s, X) \<in> FAILURES P) \<and> (\<forall>s X Y. (s, X) \<in> FAILURES P \<and> (\<forall>c. c \<in> Y \<longrightarrow> (s @ [c], {}) \<notin> FAILURES P) \<longrightarrow> (s, X \<union> Y) \<in> FAILURES P) \<and> (\<forall>s X. (s @ [tick], {}) \<in> FAILURES P \<longrightarrow> (s, X - {tick}) \<in> FAILURES P) \<and> (\<forall>s t. s \<notin> DIVERGENCES P \<or> \<not> tickFree s \<or> \<not> front_tickFree t \<or> s @ t \<in> DIVERGENCES P) \<and> (\<forall>s X. s \<notin> DIVERGENCES P \<or> (s, X) \<in> FAILURES P) \<and> (\<forall>s. s @ [tick] \<notin> DIVERGENCES P \<or> s \<in> DIVERGENCES P))"
qq_def: shows "qq = q1R * q2R + q1I * q2I"
x_def: shows "x \<equiv> \<lambda>x. 2 ^ nat \<lceil>log 2 (real (max 1 n)) - log 2 (real x)\<rceil>"
character_dc2c: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex" assumes "dcharacter n \<chi>" shows "character (residue_mult_group n) (residues_nat.dc2c n \<chi>)"
monad_zero_hrabs_iff: fixes x :: "real star" shows "(x \<in> monad 0) = (\<bar>x\<bar> \<in> monad 0)"
inverse_powr: fixes y :: "real"   and a :: "real" assumes "0 \<le> y" shows "inverse y powr a = inverse (y powr a)"
elims: fixes x :: "('c, 'd) t"   and y :: "('c \<times> 'd) list" assumes "to_list x = y" obtains xs :: "('c \<times> 'd) list" where "x = T xs" and "y = xs"
subset_zmset_def: fixes A :: "'b zmultiset"   and B :: "'b zmultiset" shows "(A \<subset>#\<^sub>z B) = (A \<subseteq>#\<^sub>z B \<and> A \<noteq> B)"
setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>pD_1: fixes p :: "'d \<times> ('e, 'f) Term.term \<times> ('e, 'f) Term.term"   and a :: "'d \<times> ('e, 'f) stateful_strand_step" assumes "p \<in> setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p a" shows "fst p = fst a"
zero_eq_add_iff_both_eq_0: fixes x :: "'d multiset"   and y :: "'d multiset" shows "({#} = x + y) = (x = {#} \<and> y = {#})"
tmmul_nb: fixes n :: "nat"   and t :: "tm"   and i :: "poly" assumes "tmbound n t" shows "tmbound n (tmmul t i)"
ik\<^sub>e\<^sub>s\<^sub>t_cons: fixes a :: "('fun, 'var) extstrand_step"   and A :: "('fun, 'var) extstrand_step list" shows "ik\<^sub>e\<^sub>s\<^sub>t (a # A) = ik\<^sub>e\<^sub>s\<^sub>t [a] \<union> ik\<^sub>e\<^sub>s\<^sub>t A"
map_pre_G\<^sub>M_def: shows "map_pre_G\<^sub>M \<equiv> \<lambda>f1 f2 f3. BNF_Composition.id_bnf \<circ> map_G0 f1 f3 \<circ> BNF_Composition.id_bnf"
arr_inv: fixes f :: "'a \<times> 'a" assumes "CC.iso f" shows "CC.arr (CC.inv f)"
Mem: fixes t1 :: "dbtm"   and t2 :: "dbtm" assumes "wf_dbtm t1"   and "wf_dbtm t2" shows "wf_dbfm (DBMem t1 t2)"
assms_1: shows "arc_to_ends G a = arc_to_ends G b"
product_dvd_fact: fixes a :: "nat"   and b :: "nat" assumes "1 < a"   and "1 < b"   and "a = b \<longrightarrow> 2 < a" shows "a * b dvd fact (a * b - 1)"
refine_IdI: fixes m :: "'a nres"   and m' :: "'a nres" assumes "m \<le> m'" shows "m \<le> \<Down> Id m'"
map\<^sub>0_def: fixes a :: "'a" shows "G.map\<^sub>0 a \<equiv> src\<^sub>D (G a)"
rComm1: fixes \<Psi>\<^sub>Q :: "'b"   and M :: "'a"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and K :: "'a"   and xvec :: "name list"   and Q' :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q \<turnstile> M \<leftrightarrow> K"   and "distinct xvec"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* N"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* xvec"   and "A\<^sub>P \<sharp>* Q'"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* K"   and "A\<^sub>Q \<sharp>* N"   and "A\<^sub>Q \<sharp>* P'"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* xvec"   and "A\<^sub>Q \<sharp>* Q'"   and "A\<^sub>Q \<sharp>* C"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* \<Psi>\<^sub>P"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* K"   and "xvec \<sharp>* Q"   and "xvec \<sharp>* \<Psi>\<^sub>Q"   and "xvec \<sharp>* C" shows "Prop (\<lparr>\<nu>*xvec\<rparr>P' \<parallel> Q')"
subsumes_accessible_globallyI: fixes s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "\<And>r\<^sub>1 \<sigma>\<^sub>1 \<tau>\<^sub>1 e\<^sub>1 r\<^sub>2 \<sigma>\<^sub>2 \<tau>\<^sub>2 e\<^sub>2. \<lbrakk>s r\<^sub>1 = Some (\<sigma>\<^sub>1, \<tau>\<^sub>1, e\<^sub>1); s r\<^sub>2 = Some (\<sigma>\<^sub>2, \<tau>\<^sub>2, e\<^sub>2)\<rbrakk> \<Longrightarrow> \<A> r\<^sub>1 r\<^sub>2 s" shows "\<A>\<^sub>G s"
aolemma: fixes xs :: "'a list"   and ys :: "'a list"   and r :: "('a \<times> 'a) set" assumes "(xs, ys) \<in> genPrefix r"   and "refl r" shows "length xs < length ys \<longrightarrow> (xs @ [ys ! length xs], ys) \<in> genPrefix r"
tys_tys'_list: fixes P :: "cld list"   and tys_tys'_list :: "(ty \<times> ty) list"   and bools :: "bool list"   and tys :: "'a list"   and tys' :: "'b list"   and ty :: "ty"   and ty' :: "ty" assumes "lift_opts (map (\<lambda>(x, y). is_sty_one P x y) tys_tys'_list) = Some bools"   and "length tys = length tys'"   and "list_all id bools"   and "(ty, ty') \<in> set tys_tys'_list" shows "is_sty_one P ty ty' = Some True"