hcju/isabelleps huggingface dataset:
assms_2: shows "y \<notin> fst ` tvsT \<tau>"
Language_init: fixes t :: "'t2" assumes "wf2 t" shows "Language2 (init2 t) = Lang2 t"
sim: shows "weak_reduction_simulation (indRelRTPO TRel) (STCal Source Target)"
relop_i: fixes t :: "t"   and \<C> :: "t_context"   and uy :: "relop_i" assumes "is_int_t t" shows "\<C> \<turnstile> [Relop_i t uy] : [t, t] _> [T_i32]"
no_loops_in_iapath: fixes V :: "'a set"   and u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a"   and a :: "'a \<times> 'a" assumes "H.gen_iapath V u p v"   and "a \<in> set p" shows "fst a \<noteq> snd a"
univ_def: fixes f :: "'b \<Rightarrow> 'a"   and X :: "'b set" shows "Equiv_Relation2.univ f X \<equiv> f (SOME x. x \<in> X)"
simps_1: fixes f :: "nat \<Rightarrow> 'a list" shows "??.Relative_Rewriting.choice f 0 = (0, 0)"
plus_mat_def: fixes A :: "'b mat"   and B :: "'b mat" shows "A + B \<equiv> mat (dim_row B) (dim_col B) (\<lambda>ij. A $$ ij + B $$ ij)"
simps_1: fixes l2 :: "'b list" shows "subset_sorted [] l2 = True"
LDLadder: shows "LeftDerivationLadder (\<alpha> @ \<delta>) D L \<gamma>"
subseteq_mset_zmsetE: fixes M :: "'b zmultiset"   and N :: "'b zmultiset" assumes "M \<subseteq>#\<^sub>z N" obtains A :: "'b multiset" and B :: "'b multiset" and C :: "'b zmultiset" where "M = zmset_of A + C" and "N = zmset_of B + C" and "A \<subseteq># B"
steps: fixes P :: "'b"   and Cal :: "'b processCalculus"   and P' :: "'b"   and Rel :: "('b \<times> 'b) set" assumes "P \<longmapsto>Cal* P'" shows "P' \<R>\<mapsto><Rel,Cal> P"
set_val_is_l_set_val: shows "l_set_val DocumentClass.type_wf set_val set_val_locs"
sub_associative: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "! (d x) \<squnion> (! (d y) \<squnion> ! (d z)) = ! (d x) \<squnion> ! (d y) \<squnion> ! (d z)"
mono_Union: fixes Retr :: "'b set \<Rightarrow> 'c set"   and Theta :: "'b set set" assumes "Bisim.mono Retr" shows "\<Union> (Retr ` Theta) \<subseteq> Retr (\<Union> Theta)"
k_mkAlgAuto_ec: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'f list"   and envTrans :: "'f \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'g"   and envObs :: "'b \<Rightarrow> 'e \<Rightarrow> 'h"   and jviewInit :: "'b \<Rightarrow> 'h \<Rightarrow> 'g"   and jviewIncr :: "'b \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> 'g"   and simf :: "'e Trace \<Rightarrow> 'i"   and simRels :: "'b \<Rightarrow> ('i \<times> 'i) set"   and simVal :: "'i \<Rightarrow> 'c \<Rightarrow> bool"   and simAbs :: "'j \<Rightarrow> 'i set"   and simObs :: "'b \<Rightarrow> 'j \<Rightarrow> 'h"   and simInit :: "'b \<Rightarrow> 'h \<Rightarrow> 'j"   and simTrans :: "'b \<Rightarrow> 'j \<Rightarrow> 'j list"   and simAction :: "'b \<Rightarrow> 'j \<Rightarrow> 'd list"   and aOps :: "('k, 'j, 'd list) MapOps"   and tOps :: "('l, 'j \<times> 'h, 'j) MapOps"   and t :: "'e Trace"   and a :: "'b" assumes "Algorithm jkbp envInit envAction envTrans envVal jview envObs jviewInit jviewIncr simf simRels simVal simAbs simObs simInit simTrans simAction aOps tOps"   and "t \<in> PreEnvironmentJView.jkbpC jkbp envInit envAction envTrans envVal jview" shows "simAbs (IncrEnvironment.runJP envObs (mkAlgAuto aOps tOps simObs simInit simTrans simAction (Algorithm.k_frontier envInit envObs simInit)) t a) = simf ` {t' \<in> PreEnvironmentJView.jkbpC jkbp envInit envAction envTrans envVal jview. jview a t' = jview a t}"
fin: shows "integral\<^sup>N M h \<noteq> \<infinity>"
ImageI: fixes a :: "'c"   and b :: "'d"   and r :: "('c \<times> 'd) set"   and A :: "'c set" assumes "(a, b) \<in> r"   and "a \<in> A" shows "b \<in> r `` A"
h2: assumes "i mod 2 = 0" shows "SQ ^ (i div 2) * real (i mod 2) * sqrt SQ + SQ ^ (i div 2) * (1 - real (i mod 2)) = sqrt SQ ^ i"
uminus_uexpr_def: fixes u :: "('a, 'b) uexpr" shows "- u = uop uminus u"
map_val_def: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and xs :: "'x" shows "map_val f xs = oalist_of_listy (map_val_raw f (list_of_oalistx xs))"
onode_ucast: fixes \<sigma> :: "nat \<Rightarrow> 'c"   and s :: "'d"   and d :: "nat"   and m :: "'e"   and \<sigma>' :: "nat \<Rightarrow> 'c"   and s' :: "'d"   and S :: "(((nat \<Rightarrow> 'c) \<times> 'd) \<times> 'e seq_action \<times> (nat \<Rightarrow> 'c) \<times> 'd) set"   and R :: "nat set"   and i :: "nat" assumes "((\<sigma>, s), unicast d m, \<sigma>', s') \<in> S"   and "d \<in> R" shows "((\<sigma>, NodeS i s R), {d}:*cast(m), \<sigma>', NodeS i s' R) \<in> onode_sos S"
simps_2: fixes K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and R :: "'b tag_const \<times> 'b tag_const \<Rightarrow> bool" shows "\<lbrakk> time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = {\<rho>. \<forall>n. R (time (Rep_run \<rho> n K\<^sub>1), time (Rep_run \<rho> n K\<^sub>2))}"
rep_eq: fixes x :: "'b interval" shows "upper x = snd (bounds_of_interval x)"
induct: fixes P :: "poly \<Rightarrow> bool"   and a0 :: "poly" assumes "\<And>c p. P p \<Longrightarrow> P (CN c 0 p)"   and "\<And>v. P (C v)"   and "\<And>v. P (Bound v)"   and "\<And>v va. P (Add v va)"   and "\<And>v va. P (Sub v va)"   and "\<And>v va. P (Mul v va)"   and "\<And>v. P (Neg v)"   and "\<And>v va. P (Pw v va)"   and "\<And>v vc vb. P (CN v (Suc vc) vb)" shows "P a0"
doesnt_read_mutually_exclusive: fixes c :: "'Com"   and x :: "'Var"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "'Com"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "doesnt_read c x"   and "\<langle>c, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>"   and "\<forall>v. \<langle>c, mds, mem(x := v)\<rangle> \<leadsto> \<langle>c', mds', mem'(x := v)\<rangle>" shows "\<not> (\<forall>v. \<langle>c, mds, mem(x := v)\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>)"
unit_dvdE: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a one" obtains c :: "'c" where "a \<noteq> zero" and "b = times a c"
avl_delete_max_2: fixes x :: "'b tree" assumes "avl x"   and "x \<noteq> ET" shows "height x = height (snd (delete_max x)) \<or> height x = height (snd (delete_max x)) + 1"
Koszul_syz_sigs_auxE: fixes v :: "'t"   and bs :: "('a \<Rightarrow>\<^sub>0 'b) list"   and k :: "nat" assumes "v \<in> set (Koszul_syz_sigs_aux bs k)" obtains i :: "nat" and j :: "nat" where "i < j" and "j < length bs" and "v = ord_term_lin.max (term_of_pair (punit.lt (bs ! i), k + j)) (term_of_pair (punit.lt (bs ! j), k + i))"
assert_def: fixes a :: "'i \<times> 'a atom"   and s :: "('i, 'a) state" shows "assert a s \<equiv> check (assert_bound a s)"
prems_3: shows "a \<in> outstanding_refs is_non_volatile_Read\<^sub>s\<^sub>b (x # sb)"
FDOM_eff_subset_FDOM_valid_states_pair: fixes a :: "('a, 'b) fmap \<times> ('a, 'c) fmap"   and PROB :: "(('a, 'b) fmap \<times> ('a, 'c) fmap) set"   and s :: "('a, 'd) fmap" assumes "a \<in> PROB"   and "s \<in> valid_states PROB" shows "fmdom' (snd a) \<subseteq> fmdom' s"
continuous_axioms_def: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set set"   and T :: "'c set set" shows "continuous_axioms f S T \<equiv> \<forall>m. m open\<^bsub>T\<^esub> \<longrightarrow> carrier\<^bsub>S\<^esub> \<inter> f -` m open\<^bsub>S\<^esub>"
2: shows "\<exists>V. Q V \<and> y \<in> T \<inter> g -` U \<and> T \<inter> g -` U \<subseteq> V \<and> V \<subseteq> W"
lincomb_cong: fixes A :: "rat vec set"   and B :: "rat vec set"   and a :: "rat vec \<Rightarrow> rat"   and b :: "rat vec \<Rightarrow> rat" assumes "A = B"   and "A \<subseteq> Rn"   and "\<And>v. v \<in> A \<Longrightarrow> a v = b v" shows "gs.lincomb a A = gs.lincomb b B"
sat_inf_imp_ground_red_fam_inter: fixes N :: "('a literal multiset \<times> label) set"   and \<iota>' :: "('a literal multiset \<times> label) inference"   and q :: "'a literal multiset set"   and \<iota> :: "'a literal multiset inference" assumes "FL.empty_ord.saturated N"   and "\<iota>' \<in> FL.Inf_from N"   and "q \<in> UNIV"   and "FL.\<G>_I_L_q q \<iota>' \<noteq> None \<and> \<iota> \<in> the (FL.\<G>_I_L_q q \<iota>')" shows "\<iota> \<in> G.Red_I q (\<Union>CL\<in>N. \<G>_F (fst CL))"
subring_degree_one_associatedI: fixes K :: "'a set"   and a :: "'a"   and a' :: "'a"   and b :: "'a" assumes "subring K R"   and "a \<in> K"   and "a' \<in> K"   and "b \<in> K"   and "a \<otimes> a' = \<one>" shows "[a, b] \<sim>\<^bsub>K [X]\<^esub> [\<one>, a' \<otimes> b]"
fmap_rel_fmdrop_fmap_rel: fixes K :: "('a \<times> 'b) set"   and A :: "('a, 'c) fmap"   and B :: "('b, 'd) fmap"   and R :: "('c \<times> 'd) set"   and p :: "'a"   and p' :: "'b" assumes "single_valued K"   and "single_valued (K\<inverse>)"   and "(A, B) \<in> \<langle>K, R\<rangle>fmap_rel"   and "(p, p') \<in> K" shows "(fmdrop p A, fmdrop p' B) \<in> \<langle>K, R\<rangle>fmap_rel"
is_processT8: fixes P :: "'a process" shows "\<forall>s X. s \<in> D P \<longrightarrow> (s, X) \<in> F P"
phull_closed_lin_red: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and A :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "phull B \<subseteq> phull A"   and "p \<in> phull A"   and "lin_red B p q" shows "q \<in> phull A"
d_add_prod_closed: fixes x :: "'a"   and y :: "'a" shows "d (d x + d y) = d x + d y"
cfs_add: fixes R :: "('c, 'd) ring_scheme"   and p :: "nat \<Rightarrow> 'c"   and q :: "nat \<Rightarrow> 'c"   and n :: "nat" assumes "UP_ring R"   and "p \<in> carrier (UP R)"   and "q \<in> carrier (UP R)" shows "(p \<oplus>\<^bsub>UP R\<^esub> q) n = p n \<oplus>\<^bsub>R\<^esub> q n"
heqr: fixes x :: "'a" assumes "x \<in> W0 \<union> T" shows "h x = r x"
isomorphicI: fixes f :: "'a" assumes "Left_a.iso f" shows "Left_a.isomorphic (Left_a.dom f) (Left_a.cod f)"
kpkki: fixes i :: "nat" assumes "i < st.d" shows "ket_plus_k k $ st.encode1 i = 1 / csqrt (2 ^ Suc k)"
tm_collapse_apps: fixes s :: "('a, 'b) tm" shows "apps (Hd (head s)) (args s) = s"
right_cancel: fixes x :: "complex vec"   and y :: "complex vec"   and z :: "complex vec" assumes "x \<in> carrier_vec d"   and "y \<in> carrier_vec d"   and "z \<in> carrier_vec d" shows "(y + x = z + x) = (y = z)"
NotS1LeadstoS6: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds"   and sigma :: "behavior" assumes "sigma \<Turnstile> \<box>ImpInv rmhist p"   and "sigma \<Turnstile> S2 rmhist p \<leadsto> S3 rmhist p"   and "sigma \<Turnstile> S3 rmhist p \<leadsto> S4 rmhist p \<or> S6 rmhist p"   and "sigma \<Turnstile> S4 rmhist p \<and> ires!p = #NotAResult \<leadsto> S4 rmhist p \<and> ires!p \<noteq> #NotAResult \<or> S5 rmhist p"   and "sigma \<Turnstile> S4 rmhist p \<and> ires!p \<noteq> #NotAResult \<leadsto> S5 rmhist p"   and "sigma \<Turnstile> S5 rmhist p \<leadsto> S6 rmhist p" shows "sigma \<Turnstile> \<not> S1 rmhist p \<leadsto> S6 rmhist p"
disc_5: shows "is_NONSAT_PUSH' NONSAT_PUSH'"
intros_2: fixes G :: "('a, 'b) cfg"   and a :: "'a"   and P :: "('a + 'b) list fset"   and w :: "'a list" assumes "in_cfls G (ffUnion ((\<lambda>\<alpha>. \<dd>\<^sub>P G \<alpha> a) |`| P)) w" shows "in_cfls G P (a # w)"
purge_Nil: fixes G :: "'a set" shows "GM_sec_model.purge G [] = []"
resumption_exhaust: fixes x :: "('f, 'g, 'h, 'i) resumption" obtains a :: "'f" where "x = Terminal a"   | b :: "'g" and r :: "('f, 'g, 'h, 'i) resumption" where "x = Linear b r"   | c :: "'h" and rs :: "'i \<Rightarrow> ('f, 'g, 'h, 'i) resumption" where "x = Branch c rs"
that: fixes q :: "'a poly"   and p :: "'a poly" assumes "constant_term q \<noteq> (0::'a)"   and "fps_of_ratfps f = fps_of_poly p / fps_of_poly q" shows "thesis"
axioms_2: 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"   and \<tau>' :: "'d \<Rightarrow> 'e" assumes "inverse_transformations A B F G \<tau> \<tau>'" shows "category B"
eval_replicate: fixes p :: "'a list"   and a :: "'a"   and n :: "nat" assumes "set p \<subseteq> carrier R"   and "a \<in> carrier R" shows "local.eval (replicate n \<zero> @ p) a = local.eval p a"
monom_uminus: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'b" shows "monom m (- a) = - monom m a"
IH: fixes l1 :: "'a list"   and h :: "'a"   and l1' :: "'a list"   and l2 :: "'a list" assumes "l1 = h # l1'"   and "l2 = l2a @ [h] @ l2b"   and "subseq l1 l2"   and "\<not> ListMem h l2a" shows "subseq l1 (h # l2b)"
cases: fixes x :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<times> 'b list" obtains P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" where "x = (P__, [])"   | P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" and xa :: "'b" and ys :: "'b list" where "x = (P, xa # ys)"
ctr_transfer_1: fixes R1 :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_fun (rel_fun (=) (rel_sp\<^sub>\<mu> R1 R2)) (rel_sp\<^sub>\<mu> R1 R2) Get Get"
antisym: fixes x :: "'d list"   and y :: "'d list" assumes "prefix x y"   and "prefix y x" shows "x = y"
degree_monic_mult: fixes p :: "'b poly"   and q :: "'b poly" assumes "monic p"   and "q \<noteq> 0" shows "degree (p * q) = degree p + degree q"
F8: shows "\<forall>w v u. word_of_int u + word_of_int v * w = word_of_int (u + v * sint w)"
adopt_node_children_subset: fixes h :: "(_) heap"   and owner_document :: "(_) document_ptr"   and node :: "(_) node_ptr"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and children' :: "(_) node_ptr list" assumes "h \<turnstile> adopt_node owner_document node \<rightarrow>\<^sub>h h'"   and "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "h' \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children'"   and "known_ptrs h"   and "type_wf h" shows "set children' \<subseteq> set children"
assms: shows "y \<cdot> x \<le> x \<cdot> (x + y)\<^sup>\<star> + y"
bijective_def: fixes R :: "('a \<times> 'b) set" shows "bijective R \<equiv> (\<forall>x y z. (x, y) \<in> R \<and> (x, z) \<in> R \<longrightarrow> y = z) \<and> (\<forall>x y z. (x, z) \<in> R \<and> (y, z) \<in> R \<longrightarrow> x = y)"
primitive_part_eq_0_iff: fixes p :: "'a poly" shows "(primitive_part p = 0) = (p = 0)"
mgu_subst_domain: fixes s :: "('b, 'a) Term.term"   and t :: "('b, 'a) Term.term"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "mgu s t = Some \<sigma>" shows "subst_domain \<sigma> \<subseteq> fv s \<union> fv t"
continuous_within: fixes x :: "'b"   and s :: "'b set"   and f :: "'b \<Rightarrow> 'd" shows "continuous (at x within s) f = (f \<longlongrightarrow> f x) (at x within s)"
cprod_infinite2': fixes r1 :: "('c \<times> 'c) set"   and r2 :: "('d \<times> 'd) set" assumes "(r1, czero) \<notin> ordIso \<and> Card_order r1"   and "Cinfinite r2"   and "r1 \<le>o r2" shows "r1 *c r2 =o r2"
simA'C: shows "similar_mat_wit A' C (four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (n - 1)) (0\<^sub>m (n - 1) 1) P') (four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (n - 1)) (0\<^sub>m (n - 1) 1) Q')"
eq_nextl_def: shows "Power.eq_nextl \<equiv> {(u, v). Power.nextl (dfa.init Power_dfa) u = Power.nextl (dfa.init Power_dfa) v}"
eq_onp_same_args: fixes P :: "'d \<Rightarrow> bool"   and x :: "'d" shows "eq_onp P x x = P x"
assms_6: fixes zs :: "('a \<times> 'c) list \<times> 'o" assumes "oalist_inv zs" shows "h zs = map_val_raw (\<lambda>k. f k (0::'b)) zs"
list_of_lmap: fixes xs :: "'a llist"   and f :: "'a \<Rightarrow> 'b" assumes "lfinite xs" shows "list_of (lmap f xs) = map f (list_of xs)"
IH_2: shows "fv\<^sub>s\<^sub>s\<^sub>t S \<subseteq> fv\<^sub>s\<^sub>e\<^sub>t (trms\<^sub>s\<^sub>s\<^sub>t S)"
f2: shows "\<forall>s sa sb w wa wb sc. \<not> low_equal s sa \<or> sb \<noteq> snd (fst (write_reg w wa wb s)) \<or> sc \<noteq> snd (fst (write_reg w wa wb sa)) \<or> low_equal sb sc"
mapCollectFilter_def: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> bool \<times> 'd"   and m :: "'e \<Rightarrow> 'f option" shows "mapCollectFilter f m = {snd (f k v) |k v. m k = Some v \<and> fst (f k v)}"
Y_ide_arr_2: fixes a :: "'c"   and g :: "'c"   and b' :: "'c"   and b :: "'c" assumes "C.ide a"   and "\<guillemotleft>g : b' \<rightarrow> b\<guillemotright>" shows "Y a g = S.mkArr (Hom.set (b, a)) (Hom.set (b', a)) (\<phi> (b', a) \<circ> (\<cdot>\<^sup>o\<^sup>p) g \<circ> \<psi> (b, a))"
FDOM_pre_subset_prob_dom_pair: fixes a :: "('d, 'e) fmap \<times> ('d, 'f) fmap"   and PROB :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set" assumes "a \<in> PROB" shows "fmdom' (fst a) \<subseteq> prob_dom PROB"
simple_imp_good_ruleset: fixes rs :: "'c rule list" assumes "simple_ruleset rs" shows "good_ruleset rs"
subst_clause_finsert: fixes \<phi> :: "'b ltln"   and \<Phi> :: "'b ltln fset"   and m :: "'b ltln \<Rightarrow> 'b ltln option" shows "subst_clause (finsert \<phi> \<Phi>) m = min_dnf (subst \<phi> m) \<otimes>\<^sub>m subst_clause \<Phi> m"
ide_in_hom: fixes a :: "'c \<times> 'c" shows "C.VV.ide a = C.VV.in_hom a a a"
assms: shows "S = {x \<in> R. x \<notin> I}"
bernoulli'_zero_iff: fixes n :: "nat" shows "(bernoulli' n = 0) = (n \<noteq> 1 \<and> odd n)"
assms_3: shows "\<forall>b\<in>basis. \<forall>(k2, \<gamma>2)\<in>one_chain2. continuous_on (path_image \<gamma>2) (\<lambda>x. F x \<bullet> b)"
combine_normalize: fixes Ks :: "'a list"   and Us :: "'a list"   and a :: "'a" assumes "set Ks \<subseteq> carrier R"   and "set Us \<subseteq> carrier R"   and "combine Ks Us = a" obtains Ks' :: "'a list" where "set (take (length Us) Ks) \<subseteq> set Ks'" and "set Ks' \<subseteq> set (take (length Us) Ks) \<union> {\<zero>}" and "length Ks' = length Us" and "combine Ks' Us = a"
useNodes_of_base: fixes g :: "'g" shows "uninst_code.ssa.useNodes_of (\<lambda>_. u_g g s1_uses g) g = uninst_code.ssa.useNodes_of (u_g g s1_uses) g"
lookup_minus: fixes f :: "'d \<Rightarrow>\<^sub>0 'c"   and g :: "'d \<Rightarrow>\<^sub>0 'c"   and k :: "'d" shows "lookup (f - g) k = lookup f k - lookup g k"
diff_self: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.group_add minus plus zero uminus" shows "minus a a = zero"
simps: fixes s :: "nat"   and l :: "cell list"   and r :: "cell list"   and lm :: "nat list"   and n :: "nat"   and ires :: "cell list" shows "mopup_jump_over2 (s, l, r) lm n ires = (\<exists>ln rn m1 m2. m1 + m2 = Suc (lm ! n) \<and> r \<noteq> [] \<and> (hd r = Oc \<longrightarrow> l = Oc \<up> m1 @ Bk \<up> ln @ Bk # Bk # ires \<and> r = Oc \<up> m2 @ Bk \<up> rn) \<and> (hd r = Bk \<longrightarrow> l = Bk \<up> ln @ Bk # ires \<and> r = Bk # Oc \<up> (m1 + m2) @ Bk \<up> rn))"
monom_mult_is_smult: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and p :: "nat \<Rightarrow> 'c" assumes "UP_ring R"   and "a \<in> carrier R"   and "p \<in> carrier (UP R)" shows "monom (UP R) a 0 \<otimes>\<^bsub>UP R\<^esub> p = a \<odot>\<^bsub>UP R\<^esub> p"
simps_1: fixes k :: "'a" shows "atoms_of_lit (k\<^sup>+) = k"
Characters_simps_3: fixes G :: "('a, 'b) monoid_scheme" shows "\<one>\<^bsub>Characters G\<^esub> = principal_char G"
no_loose_bvar_subst_bvs1'_unchanged: fixes t :: "Core.term"   and lev :: "nat"   and args :: "Core.term list" assumes "\<not> loose_bvar t lev" shows "subst_bvs1' t lev args = t"
r_def: shows "r = (if R = \<infinity> then conv_radius' + 1 else (real_of_ereal R + conv_radius') / 2)"
t: shows "\<tau> = shift (Suc (snd \<Gamma>) - k) 0 \<sigma>"
wordinterval_setminus_def: fixes r1 :: "'b wordinterval"   and r2 :: "'b wordinterval" shows "wordinterval_setminus r1 r2 = wordinterval_compress (wordinterval_setminus' r1 r2)"
fixp_induct_strong2: fixes Sup :: "'c set \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c" assumes "class.ccpo Sup less_eq less"   and "ccpo.admissible Sup less_eq P"   and "monotone less_eq less_eq f"   and "P (Sup {})"   and "\<And>x. \<lbrakk>less_eq x (ccpo.fixp Sup less_eq f); less_eq x (f x); P x\<rbrakk> \<Longrightarrow> P (f x)" shows "P (ccpo.fixp Sup less_eq f)"
tendsto_within_subset_eventuallyI: fixes g :: "'c \<Rightarrow> 'd"   and gy :: "'d"   and y :: "'c"   and Y :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and fx :: "'d"   and X :: "'c set" assumes "(g \<longlongrightarrow> gy) (at y within Y)"   and "\<forall>\<^sub>F x in at y within Y. f x = g x"   and "x = y"   and "fx = gy"   and "X - {x} \<subseteq> Y" shows "(f \<longlongrightarrow> fx) (at x within X)"
filter_empty_conv: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)"
invar_loc_axioms: shows "invar_loc G v0 D0 p D pE"
shd_def: fixes stream :: "'a stream" shows "shd stream = (case stream of x1 ## x2 \<Rightarrow> x1)"
se_fm_imp_thm: fixes A :: "fm"   and e :: "(name, hf) finfun" assumes "se_fm A"   and "ground_fm A"   and "eval_fm e A" shows "{} \<turnstile> A"
C_case_below: fixes f :: "C \<rightarrow> 'a" shows "C_case\<cdot>f \<sqsubseteq> f"
intro: fixes L :: "('a, 'b) gorder_scheme" assumes "\<And>A. A \<subseteq> carrier L \<Longrightarrow> \<exists>s. is_lub L s A"   and "\<And>A. A \<subseteq> carrier L \<Longrightarrow> \<exists>i. is_glb L i A" shows "weak_complete_lattice_axioms L"
lmap_eq_LCons_conv: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b llist"   and y :: "'a"   and ys :: "'a llist" shows "(lmap f xs = y #\<^sub>l ys) = (\<exists>x xs'. xs = x #\<^sub>l xs' \<and> y = f x \<and> ys = lmap f xs')"
lift_f_trivial_P': fixes p :: "'a freeword" assumes "p \<in> P'" shows "lift_f p = (0::'b)"
INF_cong: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b \<Rightarrow> 'c \<times> 'd"   and D :: "'b \<Rightarrow> 'c \<times> 'd" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Inf_prod (C ` A) = Inf_prod (D ` B)"
Smith_1x2_JNF_def: shows "Smith_1x2_JNF = (\<lambda>A'. let (S', Q') = Smith_1x2 (Mod_Type_Connect.to_hma\<^sub>v (Matrix.row A' 0)) in (mat_of_row (Mod_Type_Connect.from_hma\<^sub>v S'), Mod_Type_Connect.from_hma\<^sub>m Q'))"
thetaIf_Sbis: shows "thetaIf \<subseteq> Sbis"
semInt_prSubst: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" assumes "wlsSEM SEM" shows "prSubst (semInt SEM) SEM"
ifex_ite_induct2: fixes sentence :: "'a ifex \<Rightarrow> 'a ifex \<Rightarrow> 'a ifex \<Rightarrow> bool"   and i :: "'a ifex"   and t :: "'a ifex"   and e :: "'a ifex" assumes "\<And>i t e. \<lbrakk>lowest_tops [i, t, e] = None; i = Trueif\<rbrakk> \<Longrightarrow> sentence i t e"   and "\<And>i t e. \<lbrakk>lowest_tops [i, t, e] = None; i = Falseif\<rbrakk> \<Longrightarrow> sentence i t e"   and "\<And>i t e a. \<lbrakk>sentence (restrict_top i a True) (restrict_top t a True) (restrict_top e a True); sentence (restrict_top i a False) (restrict_top t a False) (restrict_top e a False); lowest_tops [i, t, e] = Some a\<rbrakk> \<Longrightarrow> sentence i t e" shows "sentence i t e"
laplace_integrand_def: fixes f :: "real \<Rightarrow> complex"   and s :: "complex"   and t :: "real" shows "laplace_integrand f s t = exp (t *\<^sub>R - s) * f t"
typs: shows "list_all (\<lambda>((idx, ty), t). typ_of t = Some ty) insts"