hcju/isabelleps huggingface dataset:
valid_path_def: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and as :: "'a list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main" shows "CFG.valid_path kind get_return_edges as \<equiv> CFG.valid_path_aux kind get_return_edges [] as"
Delta_TunerMode_CTRL: shows "Delta TunerMode_CTRL = TunerMode_CTRL_Delta"
reduce_not_nil: fixes s :: "s"   and vs :: "v list"   and es :: "e list"   and i :: "nat"   and s' :: "s"   and vs' :: "v list"   and es' :: "e list" assumes "\<lparr>s;vs;es\<rparr> \<leadsto>_ i \<lparr>s';vs';es'\<rparr>" shows "es \<noteq> []"
ys: fixes i :: "nat" assumes "i < length xs" shows "xs ! i = aform_val e (XS ! i)"
leWk: shows "W P k \<le>\<^sub>L W Q k"
bf_or_def: fixes a :: "'b \<Rightarrow> bool"   and b :: "'b \<Rightarrow> bool" shows "bf_or a b \<equiv> bf_ite a bf_True b"
paths_are_events: shows "\<forall>Q\<in>\<P>. \<forall>a\<in>Q. a \<in> \<E>"
mult_free_Abelian_group: fixes S :: "'b set"   and x :: "'b \<Rightarrow>\<^sub>0 int"   and y :: "'b \<Rightarrow>\<^sub>0 int" shows "x \<otimes>\<^bsub>free_Abelian_group S\<^esub> y = x + y"
list_emb_append_mono: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and xs' :: "'c list"   and ys :: "'c list"   and ys' :: "'c list" assumes "list_emb P xs xs'"   and "list_emb P ys ys'" shows "list_emb P (xs @ ys) (xs' @ ys')"
det_fold_sorted_set: fixes ordR :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and c' :: "'g \<Rightarrow> bool"   and f' :: "'f \<Rightarrow> 'g \<Rightarrow> 'g"   and \<sigma>' :: "'g"   and result :: "'f set \<Rightarrow> 'g"   and Rk :: "('h \<times> 'f) set"   and Rs :: "('h \<times> 'f) set \<Rightarrow> ('i \<times> 'f set) set"   and tsl :: "'i \<Rightarrow> 'h list"   and s :: "'i"   and s' :: "'f set"   and c :: "'j \<Rightarrow> bool"   and R\<sigma> :: "('j \<times> 'g) set"   and f :: "'h \<Rightarrow> 'j \<Rightarrow> 'j"   and \<sigma> :: "'j" assumes "det_fold_set ordR c' f' \<sigma>' result"   and "is_set_to_sorted_list ordR Rk Rs tsl"   and "(s, s') \<in> \<langle>Rk\<rangle>Rs"   and "(c, c') \<in> R\<sigma> \<rightarrow> bool_rel"   and "\<And>x y. \<lbrakk>(x, y) \<in> Rk; y \<in> s'\<rbrakk> \<Longrightarrow> (f x, f' y) \<in> R\<sigma> \<rightarrow> R\<sigma>"   and "(\<sigma>, \<sigma>') \<in> R\<sigma>" shows "(foldli (tsl s) c f \<sigma>, result s') \<in> R\<sigma>"
that: fixes R :: "complex \<Rightarrow> real" assumes "\<And>z. 0 < R z"   and "\<And>f. \<lbrakk>f holomorphic_on cball 0 (R (f 0)); \<And>z. cmod z \<le> R (f 0) \<Longrightarrow> f z \<noteq> 0 \<and> f z \<noteq> 1\<rbrakk> \<Longrightarrow> cmod (deriv f 0) < 1" shows "thesis"
diagonal_to_Smith_i_dvd_ii: fixes bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and xs :: "nat list"   and A :: "(('a, 'b) vec, 'c) vec"   and i :: "nat" assumes "is_bezout_ext bezout" shows "diagonal_to_Smith_i xs A i bezout $ from_nat i $ from_nat i dvd A $ from_nat i $ from_nat i"
max_in_subcomplex: fixes Y :: "'a set set"   and y :: "'a set" assumes "order.greater_eq X Y \<and> SimplicialComplex Y"   and "y \<in> Y"   and "maxsimp y" shows "SimplicialComplex.maxsimp Y y"
composite_pseudofunctor_axioms: shows "composite_pseudofunctor (\<cdot>\<^sub>B) (\<star>\<^sub>B) \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F H \<Phi>\<^sub>H"
inf_matrix_def: fixes f :: "'c \<times> 'c \<Rightarrow> 'd"   and g :: "'c \<times> 'c \<Rightarrow> 'd" shows "f \<otimes> g = (\<lambda>e. f e \<sqinter> g e)"
abs_eq: fixes x :: "'b \<times> 'b" assumes "fractrel x x" shows "inverse (abs_fract x) = abs_fract (if fst x = (0::'b) then (0::'b, 1::'b) else (snd x, fst x))"
lit_ueval: fixes x :: "'e"   and b :: "'f" shows "\<lbrakk>\<guillemotleft>x\<guillemotright>\<rbrakk>\<^sub>e b = x"
sum_height_measure_is_OV2: fixes l :: "nat \<Rightarrow> nat" shows "sum_heights_measure l \<le> 2 * (card V)\<^sup>2"
exhaust: fixes y :: "('c, 'a, 'b) namespace" obtains x1 :: "('a \<times> 'b) list" and x2 :: "('c \<times> ('c, 'a, 'b) namespace) list" where "y = Bind x1 x2"
inv_comm: fixes x :: "'a"   and y :: "'a" assumes "x \<oplus> y = \<zero>"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "y \<oplus> x = \<zero>"
lists_no_head_sub: fixes b :: "'a list"   and us :: "'a list list" assumes "b \<in> \<C>"   and "us \<in> lists (no_head_gen \<C> b)" shows "us \<in> lists \<langle>\<C>\<rangle>"
zero_in_factorization_iff: fixes fctrs :: "'b \<times> ('b \<times> nat) list"   and p :: "'b poly" assumes "is_factorization_of fctrs p" shows "(constant_term p = (0::'b)) = (p = 0 \<or> (0::'b) \<in> fst ` set (snd fctrs))"
irreflexive_subset: fixes B :: "'a set" assumes "B \<subseteq> A" shows "irreflexive B (\<sqsubset>)"
the_equality: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "P a"   and "\<And>x. P x \<Longrightarrow> x = a" shows "(THE x. P x) = a"
o4: fixes a :: "name"   and s1 :: "lam"   and s2 :: "lam"   and t1 :: "lam"   and t2 :: "lam" assumes "a \<sharp> (s1, s2)"   and " s1 \<longrightarrow>\<^sub>1 s2"   and " t1 \<longrightarrow>\<^sub>1 t2" shows " App (Lam [a].t1) s1 \<longrightarrow>\<^sub>1 t2[a::=s2]"
sets_S_eq: fixes x :: "'a" shows "sets S = sigma_sets UNIV {{t..} \<times> A |t A. A \<subseteq> - I x \<or> (\<exists>s\<in>I x. A = {s})}"
induct: fixes P :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool"   and a0 :: "'a word"   and a1 :: "'a word" assumes "\<And>a b. (a \<noteq> b \<Longrightarrow> P (a + 1) b) \<Longrightarrow> P a b" shows "P a0 a1"
b_disjoint: fixes K :: "nat set"   and j0 :: "nat"   and i :: "nat"   and j :: "nat"   and i' :: "nat" assumes "K \<subseteq> {j0<..}"   and "finite K"   and "j0 \<le> card K"   and "i < j"   and "i' < j"   and "i \<noteq> i'"   and "j \<le> j0" shows "b (enum K i) (j0, i) \<inter> b (enum K i') (j0, i') = {}"
smooth_on_compose: fixes k :: "enat"   and T :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and S :: "'e set"   and g :: "'e \<Rightarrow> 'a" assumes "k-smooth_on T f"   and "k-smooth_on S g"   and "open S"   and "open T"   and "g ` S \<subseteq> T" shows "k-smooth_on S (f \<circ> g)"
scaleR_zero_right: fixes a :: "real" shows "a *\<^sub>R (0::'a) = (0::'a)"
comp_inv_arr: fixes f :: "'b"   and g :: "'b" assumes "B.inverse_arrows f g" shows "B g f = B.dom f"
sequence_of_function_of: fixes enum :: "nat \<Rightarrow> 'b"   and A :: "'b set"   and xs :: "'c list"   and B :: "'c set" assumes "bij_betw enum {0..<card A} A"   and "set xs \<subseteq> B"   and "length xs = card A" shows "sequence_of A enum (function_of A enum xs) = xs"
outside_no_overlap: fixes S :: "'a set" shows "outside S \<inter> S = {}"
set_of_graph_rulesD_1: fixes Rs :: "(('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph) set"   and R :: "('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph" assumes "set_of_graph_rules Rs"   and "R \<in> Rs" shows "finite_graph (fst R)"
le_comp: 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"   and y :: "'b"   and z :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "less_eq x y" shows "less_eq (times z x) (times z y)"
Aheap_simp2: fixes x :: "Vars.var"   and e' :: "exp"   and e :: "exp"   and a :: "Arity" assumes "x \<notin> fv e'" shows "Aheap [(x, e')] e\<cdot>a = Aheap_nonrec x e'\<cdot>(Aexp_syn' a e, ccExp_syn a e)"
r_step_equiv_step: fixes c :: "(recf \<times> nat list \<times> nat list) list \<times> nat option" assumes "valid (fst c)" shows "eval r_step [encode_config c] \<down>= encode_config (step c)"
broadcast_is_valid: fixes m :: "'a \<times> ('a, 'b) operation"   and i :: "nat" assumes "Broadcast m \<in> set (history i)" shows "\<exists>state. valid_behaviours state m"
p: fixes s :: "'s" assumes "guard s"   and "I s" shows "p \<le> spmf (map_spmf guard (body s)) False"
sym_def: fixes a1 :: "'a"   and a0 :: "'a" shows "\
[a1, a0] \<equiv> if ide a0 \<and> ide a1 then \<langle>\<pp>\<^sub>0[a1, a0], \<pp>\<^sub>1[a1, a0]\<rangle> else null"
path_component_eq_eq: fixes S :: "'c set"   and x :: "'c"   and y :: "'c" shows "(path_component S x = path_component S y) = (x \<notin> S \<and> y \<notin> S \<or> x \<in> S \<and> y \<in> S \<and> path_component S x y)"
is_inf_meet: fixes x :: "'c"   and y :: "'c" shows "is_inf x y (x \<sqinter> y)"
spr_ii_simVal_def: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd \<times> 'e) GC list"   and agents :: "'b list"   and envVal :: "('b, 'f, 'g, 'd, 'e) BEState \<Rightarrow> 'c \<Rightarrow> bool" assumes "FiniteBroadcastEnvironmentIndependentInit jkbp agents" shows "FiniteBroadcastEnvironmentIndependentInit.spr_ii_simVal envVal \<equiv> envVal \<circ> snd"
fresh_swap_id: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort"   and x1 :: "'var"   and x2 :: "'var" assumes "good X"   and "fresh xs x1 X"   and "fresh xs x2 X" shows "X #[x1 \<and> x2]_xs = X"
kripkeI: fixes M :: "('e, 'g, 'f) KripkeStructure" assumes "\<And>a. relations M a \<subseteq> worlds M \<times> worlds M" shows "kripke M"
left_cancel: fixes a :: "'a perm"   and b :: "'a perm"   and c :: "'a perm" shows "(a * b = a * c) = (b = c)"
monom_mult_in_pmdl: fixes b :: "'c \<Rightarrow>\<^sub>0 'd"   and B :: "('c \<Rightarrow>\<^sub>0 'd) set"   and c :: "'d"   and t :: "'c" assumes "b \<in> B" shows "MPoly_Type_Class.punit.monom_mult c t b \<in> MPoly_Type_Class.punit.pmdl B"
agree_restrict_state_succ_idempot: fixes a :: "('d, 'e) fmap \<times> ('d, 'e) fmap"   and PROB :: "(('d, 'e) fmap \<times> ('d, 'e) fmap) set"   and s :: "('d, 'e) fmap"   and vs :: "'d set" assumes "a \<in> PROB"   and "s \<in> valid_states PROB"   and "agree (fmrestrict_set vs (snd a)) (fmrestrict_set vs s)" shows "fmrestrict_set vs (state_succ s a) = fmrestrict_set vs s"
minus_of_nat: fixes x :: "'a"   and n :: "nat" shows "f (gm x (of_nat n)) = f x"
atLeastAtMostSuc_conv: fixes m :: "nat"   and n :: "nat" assumes "m \<le> Suc n" shows "{m..Suc n} = insert (Suc n) {m..n}"
union_ac_3: fixes M :: "'b multiset"   and N :: "'b multiset"   and K :: "'b multiset" shows "M + (N + K) = N + (M + K)"
sconcat_strict: shows "sconcat\<cdot>\<bottom> = \<bottom>"
eval_fds_eqI: fixes f :: "'a fds"   and s :: "'a"   and L :: "'a" assumes "(\<lambda>n. fds_nth f (Suc n) / nat_power (Suc n) s) sums L" shows "eval_fds f s = L"
simps: fixes a1 :: "'a trm"   and a2 :: "'a trm" shows "a1 >>\<^sup>* a2 = ((\<exists>A. a1 = A \<and> a2 = A) \<or> (\<exists>A A' A''. a1 = A \<and> a2 = A'' \<and> A >>\<^sup>* A' \<and> A' >> A''))"
mult_poly_p_add_mset_same: fixes q' :: "(char list multiset \<times> int) multiset"   and A :: "(char list multiset \<times> int) multiset"   and r :: "(char list multiset \<times> int) multiset"   and B :: "(char list multiset \<times> int) multiset"   and r' :: "(char list multiset \<times> int) multiset"   and x :: "char list multiset \<times> int" assumes "(mult_poly_p q')\<^sup>*\<^sup>* (A, r) (B, r')" shows "(mult_poly_p q')\<^sup>*\<^sup>* (add_mset x A, r) (add_mset x B, r')"
chain_mono_less: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Y :: "nat \<Rightarrow> 'c"   and i :: "nat"   and j :: "nat" assumes "class.po below"   and "po.chain below Y"   and "i < j" shows "below (Y i) (Y j)"
Inf_from_def: fixes N :: "'a literal multiset set" shows "F.Inf_from N = {\<iota> \<in> F_Inf. set (prems_of \<iota>) \<subseteq> N}"
prems_1: shows "lookup2 (h1 # t1) l2 f = Some e"
0: shows "x \<notin> set_tree B \<and> x \<notin> set_tree A"
monom_Suc_1: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R" shows "monom P a (Suc n) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P a n"
of_alist_def: fixes xs :: "('b \<times> 'c) list" shows "of_alist xs = foldr (\<lambda>(k, v). update k v) xs Mapping.empty"
hh_2: shows "MP_Rel H (of_int_poly (Mp H))"
inject_1: fixes x1 :: "'e"   and y1 :: "'e" shows "(pure x1 = pure y1) = (x1 = y1)"
map\<^sub>0_simps_3: fixes a :: "'c" assumes "C.obj a" shows "trg\<^sub>C (\<epsilon>\<^sub>0 a) = I\<^sub>C.map\<^sub>0 a"
closed_sum_cspan: fixes X :: "'b set"   and Y :: "'b set" shows "cspan X +\<^sub>M cspan Y = closure (cspan (X \<union> Y))"
eq_Max_iff: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and A :: "'g set"   and m :: "'g" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "(m = linorder.Max less_eq A) = (m \<in> A \<and> (\<forall>a\<in>A. less_eq a m))"
subseqs_powset: fixes xs :: "'b list" shows "set ` ssubseqs xs = Pow (set xs)"
eventually_def: fixes P :: "'b \<Rightarrow> bool"   and F :: "'b filter" shows "Filter.eventually P F = Rep_filter F P"
shiftl_shiftl: fixes a :: "'b word"   and b :: "nat"   and c :: "nat" shows "a << b << c = a << b + c"
deepR_list: fixes n :: "nat"   and m :: "('a, 'b) FingerTreeStruc"   and pr :: "('a, 'b) Digit"   and sf :: "('a, 'b) Node list" assumes "FingerTreeStruc.is_leveln_ftree (Suc n) m \<and> FingerTreeStruc.is_measured_ftree m"   and "FingerTreeStruc.is_leveln_digit n pr \<and> FingerTreeStruc.is_measured_digit pr"   and "\<forall>x\<in>set sf. (FingerTreeStruc.is_measured_node x \<and> FingerTreeStruc.is_leveln_node n x) \<and> length sf \<le> 4" shows "FingerTreeStruc.toList (FingerTreeStruc.deepR pr m sf) = FingerTreeStruc.digitToList pr @ FingerTreeStruc.toList m @ FingerTreeStruc.nlistToList sf"
p1: fixes l :: "nat" shows "tensor_P mat_O (1\<^sub>m K) * tensor_P (proj_psi'_l l) (proj_k l) * tensor_P mat_O (1\<^sub>m K) = tensor_P (proj_psi_l l) (proj_k l)"
neval_Suc_simp: fixes x :: "((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat)"   and y :: "((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat)" shows "conc.neval x (Suc 0) y = eval_abv\<^sub>C x y"
group_BijGroup: fixes S :: "'d set" shows "Group.group (BijGroup S)"
dijkstra_loop_def: shows "dijkstra_loop \<equiv> while (\<lambda>(Q, V). \<not> Q_is_empty Q) (\<lambda>(Q, V). let (u, du) = Q_getmin Q; Q = Q_relax_outgoing u du V Q; Q = Q_delete u Q in Let (M_update u du V) (Pair Q)) (Q_initQ, M_initV)"
collect_clkvt_def: fixes A :: "('b \<times> ('a, 'd) cconstraint \<times> ('a set \<times> 'b) pmf) set \<times> ('b \<Rightarrow> ('a, 'd) cconstraint)" shows "PTA.collect_clkvt A = \<Union> ((fst \<circ> snd \<circ> snd \<circ> snd) ` Edges A)"
DJ: shows "dom (\<alpha> m1) \<inter> dom (\<alpha> m2) = {}"
simps: fixes S :: "'a set"   and I :: "'a set" shows "states (Kripke S I) = S"
Phi_def: fixes as :: "(nat \<times> nat list) list"   and rs :: "'b list"   and s0 :: "'b list"   and n :: "nat" assumes "MTF_Off as rs s0" shows "MTF_Off.Phi as rs s0 n = int (card (Inv (MTF_Off.s_A as rs s0 n) (MTF_Off.s_mtf rs s0 n)))"
set_tl_append': fixes xs :: "'a list"   and ys :: "'a list" shows "set (tl (xs @ ys)) \<subseteq> set (tl xs) \<union> set ys"
distinct_27: fixes x31 :: "nat set"   and x32 :: "nat set"   and x33 :: "'a"   and x61 :: "nat"   and x62 :: "nat"   and x63 :: "nat" shows "x31\<not>x32:arrive(x33) \<noteq> x61:newpkt(x62, x63)"
removeAt_def: fixes n :: "nat"   and ls :: "'a list" shows "removeAt n ls = take n ls @ drop (n + 1) ls"
beta: fixes A :: "'a trm"   and A' :: "'a trm"   and B :: "'a trm"   and B' :: "'a trm"   and x :: "'a"   and T :: "PreSimplyTyped.type" assumes "A >>> A'"   and "B >>> B'" shows "App (Fn x T A) B >>> A'[x ::= B']"
binsert_set: fixes h :: "'b \<Rightarrow> int"   and t :: "'b Tree"   and e :: "'b" shows "sortedTree h t \<longrightarrow> setOf (binsert h e t) = setOf t - eqs h e \<union> {e}"
exi_mono: fixes A :: "fmla"   and H :: "fmla set"   and B :: "fmla"   and i :: "name" assumes "insert A H \<turnstile> B"   and "\<forall>C\<in>H. atom i \<sharp> C" shows "insert (exi i A) H \<turnstile> exi i B"
prod_result_cases: fixes r :: "'b \<times> ('c, 'd) result" obtains s :: "'b" and v :: "'c" where "r = (sb__, Rval v)"   | s :: "'b" and err :: "'d error_result" where "r = (s, Rerr err)"
leg1_simps_3: shows "src r\<^sub>0s\<^sub>1.p\<^sub>1 = src r\<^sub>0s\<^sub>1.p\<^sub>0"
Sigma: fixes A :: "'a set"   and B :: "'a \<Rightarrow> 'b set"   and g :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "finite A"   and "\<forall>x\<in>A. finite (B x)" shows "(\<Prod>x\<in>A. prod (g x) (B x)) = (\<Prod>(x, y)\<in>Sigma A B. g x y)"
r_delay: fixes i :: "nat"   and x :: "nat" shows "eval r_delay [i, x] \<down>= (if \<exists>j<e_length x. 0 < the (eval r_result1 [e_length x, i, e_take (Suc j) x]) then Suc (GREATEST j. j < e_length x \<and> 0 < the (eval r_result1 [e_length x, i, e_take (Suc j) x])) else 0)"
startfin_4: shows "(a $ 1 \<le> pathfinish g $ 1 \<and> pathfinish g $ 1 \<le> b $ 1) \<and> a $ 2 \<le> pathfinish g $ 2 \<and> pathfinish g $ 2 \<le> b $ 2"
simps_1: fixes c :: "'d" shows "ODE_dom (OVar c) = {}"
Nset_pre1: fixes n :: "nat" shows "{i. i \<le> Suc n} - {Suc n} = {i. i \<le> n}"
pr1_def: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "hfsetcat.pr1 a b = (if ide a \<and> ide b then mkArr (local.set (hfsetcat.prod a b)) (local.set a) (\<lambda>x. UP (hfst (DOWN x))) else null)"
consistent_witness_list: fixes A :: "'c set"   and S :: "(('d, 'c) fm list \<times> 'c) set"   and ps :: "('d, 'c) fm list"   and a :: "'c"   and used :: "'c set" assumes "infinite UNIV"   and "consistent A S"   and "(ps, a) \<in> S"   and "finite used"   and "A \<union> \<Union> (block_nominals ` S) \<subseteq> used" shows "consistent A ({(witness_list ps used, a)} \<union> S)"
prime_dvd_mult_iff: 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 p :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p" shows "dvd.dvd times p (times a b) = (dvd.dvd times p a \<or> dvd.dvd times p b)"
continuous_at_avoid: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'b"   and a :: "'b" assumes "isCont f x"   and "f x \<noteq> a" shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
wf_subtype_env: fixes \<Gamma> :: "binding list"   and P :: "POPLmarkRecord.type"   and Q :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile> P <: Q" shows "\<Gamma> \<turnstile>\<^sub>w\<^sub>f"
mem_Collect_eq: fixes a :: "'a"   and P :: "'a \<Rightarrow> bool" shows "((a::'a) \<in> Collect (P::'a \<Rightarrow> bool)) = P a"
cod_eqI': fixes a :: "'c"   and f :: "'c" assumes "a \<in> C.codomains f" shows "a = C.cod f"
memberdecl_is_static_mdecl_simp: fixes m :: "mdecl" shows "is_static (mdecl m) = is_static m"
cint: shows "((\<lambda>z. f (- z)) has_contour_integral - i) \<gamma>"
sf_def: fixes x :: "nat" shows "sf x = x * (x + 1) div 2"
simps_3: shows "nnf (Neg TrueF) = FalseF"
in_range_empty: fixes h :: "heap" shows "in_range (h, {})"
cexpr_sem_marg_dens2: fixes vs :: "nat list"   and vs' :: "nat list"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "cexpr"   and x :: "nat"   and y :: "nat"   and \<rho> :: "nat \<Rightarrow> val" assumes "cdens_ctxt_invar vs vs' \<Gamma> \<delta>"   and "x \<in> set vs"   and "y \<in> set vs"   and "x \<noteq> y"   and "\<rho> \<in> space (state_measure (set vs') \<Gamma>)" shows "AE z in stock_measure (PRODUCT (\<Gamma> x) (\<Gamma> y)). ennreal (extract_real (cexpr_sem (case_nat z \<rho>) (marg_dens2_cexpr \<Gamma> vs x y \<delta>))) = marg_dens2 (dens_ctxt_\<alpha> (vs, vs', \<Gamma>, \<delta>)) x y \<rho> z"
thetaWhileZOT_def: shows "thetaWhileZOT \<equiv> {(while tst do c, while tst do d) |tst c d. compatTst tst \<and> c \<approx>01T d} \<union> {(c1 ;; while tst do c, d1 ;; while tst do d) |tst c1 d1 c d. compatTst tst \<and> c1 \<approx>01T d1 \<and> c \<approx>01T d}"
inj_onI: fixes A :: "'q set"   and f :: "'q \<Rightarrow> 'r" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; f x = f y\<rbrakk> \<Longrightarrow> x = y" shows "inj_on f A"
is_natural_transformation: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e" assumes "inverse_transformation A B F G \<tau>" shows "natural_transformation A B G F (inverse_transformation.map A B G \<tau>)"
hyps_2: shows "vo = (\<lambda>k. if k = 0 then None else Some (SOME c. v c = k))"
allvars_finite: shows "finite Denotational_Semantics.allvars"
pochhammer_Suc_prod: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b"   and n :: "nat" assumes "class.comm_semiring_1 times one plus zero" shows "comm_semiring_1.pochhammer times one plus zero a (Suc n) = comm_monoid_mult.prod times one (\<lambda>i. plus a (semiring_1.of_nat one plus zero i)) {0..n}"
Abs_io_inject: fixes x :: "\<^url> \<Rightarrow> 'c \<times> \<^url>"   and y :: "\<^url> \<Rightarrow> 'c \<times> \<^url>" assumes "x \<in> UNIV"   and "y \<in> UNIV" shows "(Abs_io x = Abs_io y) = (x = y)"
sup_pp_semi_commute: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" shows "x \<oplus> \<ominus> \<ominus> y \<preceq> \<ominus> \<ominus> (x \<oplus> y)"
bin_nth_cat: fixes k :: "nat"   and y :: "int"   and x :: "int"   and n :: "nat" shows "bit (concat_bit k y x) n = (if n < k then bit y n else bit x (n - k))"
simps_2: fixes xs :: "'c list" shows "C_merge_adj [xs] = 0"
more_than_one: shows "\<nexists>m. obs_intra (sourcenode a) \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub> = {m} \<or> obs_intra (sourcenode a) \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub> = {}"
downset_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "ord.downset less_eq = ord.downset_set less_eq \<circ> \<eta>"
qmpt_density: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and h :: "'b \<Rightarrow> ennreal" assumes "qmpt M T"   and "h \<in> borel_measurable M"   and "AE x in M. h x \<noteq> 0"   and "AE x in M. h x \<noteq> \<infinity>" shows "qmpt (density M h) T"
in_homI: fixes f :: "'t"   and a :: "'t"   and b :: "'t" assumes "S'.arr f"   and "S'.dom f = a"   and "S'.cod f = b" shows "\<guillemotleft>f : a \<rightarrow>' b\<guillemotright>"
infinite_iff_countable_subset: fixes S :: "'b set" shows "infinite S = (\<exists>f. inj f \<and> Set.range f \<subseteq> S)"
eventually_floor_less: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'b"   and F :: "'a filter" assumes "(f \<longlongrightarrow> l) F"   and "l \<notin> \<int>" shows "\<forall>\<^sub>F x in F. of_int \<lfloor>l\<rfloor> < f x"
lfinite_llist_of_stream: fixes xs :: "'c stream" shows "\<not> lfinite (llist_of_stream xs)"
tensor_mat_adjoint: fixes m1 :: "'a mat"   and m2 :: "'a mat" assumes "m1 \<in> carrier_mat d1 d1"   and "m2 \<in> carrier_mat d2 d2" shows "adjoint (tensor_mat m1 m2) = tensor_mat (adjoint m1) (adjoint m2)"
imp: shows "AE x\<in>A in M. x \<in> B"
c: shows "as\<^sub>1 = map (\<lambda>v. (v, True)) (add_effects_of op)"