hcju/isabelleps huggingface dataset:
add_comm: fixes x :: "'a ppath set"   and y :: "'a ppath set" shows "x \<union> y = y \<union> x"
preserves_seq: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "B.VV.seq g f" shows "B.seq (fst g \<star>\<^sub>B snd g) (fst f \<star>\<^sub>B snd f)"
injectivity_2: fixes B :: "'b set"   and C :: "'b set"   and P :: "'b set set"   and j :: "nat"   and P' :: "'b set set"   and j' :: "nat"   and B' :: "'b set"   and k :: "nat"   and B'' :: "'b set"   and k' :: "nat"   and Q :: "'b set set"   and Q' :: "'b set set"   and f :: "'b \<Rightarrow> 'b set"   and g :: "'b \<Rightarrow> 'b set"   and P'' :: "'b set set"   and P''' :: "'b set set" assumes "B \<inter> C = {}"   and "(partition_on C P \<and> card P = j) \<and> partition_on C P' \<and> card P' = j'"   and "(B' \<subseteq> B \<and> card B' = k) \<and> B'' \<subseteq> B \<and> card B'' = k'"   and "partition_on B' Q \<and> partition_on B'' Q'"   and "f \<in> B - B' \<rightarrow>\<^sub>E P \<and> g \<in> B - B'' \<rightarrow>\<^sub>E P'"   and "P'' = (\<lambda>X. X \<union> {x \<in> B - B'. f x = X}) ` P \<and> P''' = (\<lambda>X. X \<union> {x \<in> B - B''. g x = X}) ` P'"   and "P'' \<union> Q = P''' \<union> Q'" shows "Q = Q'"
sine_law_triangle': fixes b :: "'b"   and c :: "'b"   and a :: "'b" assumes "b \<noteq> c" shows "sin (angle a b c) = sin (angle b a c) * dist a c / dist b c"
assms_1: shows "\<forall>M\<in>Chains leq. finite M"
assms_2: shows "set (coeffs q) \<subseteq> \<real>"
reach: shows "\<Gamma>\<turnstile> (c, s) \<rightarrow>\<^sup>* (c1, s1)"
elims_2: fixes x :: "'b heap" assumes "is_root x" obtains "x = heap.Empty"   | xa :: "'b" and hs :: "'b heap list" where "x = Hp xa hs" and "no_Emptys hs"
assms: shows "equivalent_bicategories (\<bullet>) (\<star>) assoc unit src trg V\<^sub>1 H\<^sub>1 \<a>\<^sub>1 \<i>\<^sub>1 src\<^sub>1 trg\<^sub>1"
funToCone_mapsto: shows "funToCone \<in> Pi I (S.set \<circ> D) \<inter> S.extensional' I \<rightarrow> cones S.unity"
finite_induct: fixes F :: "'q set"   and P :: "'q set \<Rightarrow> bool" assumes "finite F"   and "P {}"   and "\<And>x F. \<lbrakk>finite F; x \<notin> F; P F\<rbrakk> \<Longrightarrow> P (insert x F)" shows "P F"
inject_1: fixes x1 :: "'e"   and y1 :: "'e" shows "(Inl x1 = Inl y1) = (x1 = y1)"
chain_def: fixes C :: "'a set" shows "chain_on P C {x \<in> A. \<not> Q x} = (C \<subseteq> {x \<in> A. \<not> Q x} \<and> (\<forall>x\<in>C. \<forall>y\<in>C. x \<sqsubseteq> y \<or> y \<sqsubseteq> x))"
Max_ge: fixes A :: "'b Word.word set"   and x :: "'b Word.word" assumes "finite A"   and "x \<in> A" shows "x \<le>s signed.Max A"
neg_scaleR: fixes x :: "real"   and b :: "real \<times> real"   and c :: "real \<times> real" assumes "x < 0" shows "ccw' 0 (x *\<^sub>R b) c = ccw' 0 c b"
fps_exp_compose_linear: fixes d :: "'a"   and c :: "'a" shows "fps_exp d oo fps_const c * fps_X = fps_exp (c * d)"
Hermite_ring_class_def: shows "OFCLASS('b, Hermite_ring_class) \<equiv> (OFCLASS('b, comm_ring_1_class) &&& class.Hermite_ring TYPE('b))"
simps_1: fixes uu :: "'effect"   and uv :: "'effect fs_set"   and P :: "'state"   and \<phi> :: "'pred" shows "AC (uu, uv, P) \<turnstile>\<^sub>L \<phi> = P \<turnstile> \<phi>"
CD_on_inj_on_Xd: fixes ds :: "'b set"   and X :: "'a set" shows "inj_on Xd (CD_on ds X)"
c_2: shows "\<bar>lead_coeff f\<bar> = d"
Fvars_\<phi>L: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "Fvars (\<phi>L \<phi>) = {}"
assms: shows "subgraph G\<^sub>1 G\<^sub>2"
HOLCF_trans_rules_1: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> z" shows "x \<sqsubseteq> z"
ipurge_ref_distrib_inter: fixes I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and u :: "'d"   and xs :: "'c list"   and X :: "'c set"   and Y :: "'c set" shows "ipurge_ref I D u xs (X \<inter> Y) = ipurge_ref I D u xs X \<inter> ipurge_ref I D u xs Y"
pred_stream_iff: fixes P :: "'a \<Rightarrow> bool"   and s :: "'a stream" shows "pred_stream P s = Ball (sset s) P"
eint_pos_times_is_pos: fixes c :: "eint"   and b :: "eint" assumes "0 < c"   and "0 < b" shows "0 < b * c"
igSwapAllIPresIGWlsAll_def: fixes MOD :: "('b, 'c, 'varSort, 'sort, 'd, 'e, 'f, 'g, 'h) model_scheme" shows "igSwapAllIPresIGWlsAll MOD \<equiv> igSwapIPresIGWls MOD \<and> igSwapAbsIPresIGWlsAbs MOD"
iso: shows "i \<in> Group.iso \<F>\<^bsub>X\<^esub> \<F>\<^bsub>Y\<^esub>"
relation_in_carrier: fixes outcomes :: "'a set"   and \<R> :: "('a pmf \<times> 'a pmf) set"   and x :: "'a pmf"   and y :: "'a pmf" assumes "rational_preference (\<P> outcomes) \<R>"   and "independent_vnm (\<P> outcomes) \<R>"   and "x \<succeq>[\<R>] y" shows "x \<in> \<P> outcomes \<and> y \<in> \<P> outcomes"
simps: fixes a1 :: "'b llist"   and a2 :: "'b llist" shows "emb a1 a2 = ((\<exists>xs. a1 = LNil \<and> a2 = xs \<and> lfinite xs) \<or> (\<exists>xs ys x zs. a1 = LCons x xs \<and> a2 = prepend zs (LCons x ys) \<and> emb xs ys))"
isSimplePath_append: fixes s :: "nat"   and p1 :: "(nat \<times> nat) list"   and p2 :: "(nat \<times> nat) list"   and t :: "nat" shows "cf.isSimplePath s (p1 @ p2) t = (\<exists>u. cf.isSimplePath s p1 u \<and> cf.isSimplePath u p2 t \<and> set (cf.pathVertices_fwd s p1) \<inter> set (cf.pathVertices_fwd u p2) = {u})"
exhaust: fixes y :: "'g + 'h" obtains a :: "'g" where "y = Inl a"   | b :: "'h" where "y = Inr b"
gets_to_return: fixes v :: "'c" shows "gets (\<lambda>s. v) = return v"
OF: fixes i :: "'b" assumes "i \<in> I" shows "ofilter r (A i)"
substBinp_def2: fixes ys :: "'varSort"   and Y :: "('b, 'c, 'varSort, 'var, 'd) qTerm set"   and y :: "'var" shows "substBinp ys Y y = lift (substAbs ys Y y)"
zero_diff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and a :: "'d" assumes "class.comm_monoid_diff plus minus zero" shows "minus zero a = zero"
HComplex_add_hcomplex_of_hypreal: fixes x :: "real star"   and y :: "real star"   and r :: "real star" shows "HComplex x y + hcomplex_of_hypreal r = HComplex (x + r) y"
w_def: shows "w = geodesic_segment_param Gxy x (Gromov_product_at x y z)"
basis_def: fixes A :: "'a vec set" shows "KB.basis A = (KB.lin_indpt A \<and> KB.span A = mat_kernel B \<and> A \<subseteq> mat_kernel B)"
append_eq_appendI: fixes xs :: "'f list"   and xs1 :: "'f list"   and zs :: "'f list"   and ys :: "'f list"   and us :: "'f list" assumes "xs @ xs1 = zs"   and "ys = xs1 @ us" shows "xs @ ys = zs @ us"
leg1_in_hom_2: shows "\<guillemotleft>t\<^sub>1 : t\<^sub>1 \<Rightarrow> t\<^sub>1\<guillemotright>"
if_bisimJ2JVM_start: fixes C :: "String.literal"   and M :: "String.literal"   and vs :: "'addr val list" assumes "wf_J_prog P"   and "wf_start_state P C M vs" shows "if_bisimJ2JVM (init_fin_lift_state Running (start_state J_local_start P C M vs)) (init_fin_lift_state Running (start_state JVM_local_start (J2JVM P) C M vs))"
sphere_translation: fixes a :: "'c"   and c :: "'c"   and r :: "real" shows "sphere (a + c) r = (+) a ` sphere c r"
subdomainI: fixes H :: "'a set" assumes "subcring H R"   and "\<one> \<noteq> \<zero>"   and "\<And>h1 h2. \<lbrakk>h1 \<in> H; h2 \<in> H; h1 \<otimes> h2 = \<zero>\<rbrakk> \<Longrightarrow> h1 = \<zero> \<or> h2 = \<zero>" shows "subdomain H R"
replicate_length_filter: fixes x :: "'k"   and xs :: "'k list" shows "replicate (length (filter ((=) x) xs)) x = filter ((=) x) xs"
bij_betw_cong: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'f"   and A' :: "'f set" assumes "\<And>a. a \<in> A \<Longrightarrow> f a = g a" shows "bij_betw f A A' = bij_betw g A A'"
avg_proj_is_proj_right: fixes W :: "'v set"   and U :: "'v set"   and v :: "'v" assumes "VectorSpace (\<sharp>\<cdot>) W"   and "RModule FG (\<cdot>) U \<and> U \<subseteq> V"   and "add_independentS [W, U]"   and "V = W \<oplus> U"   and "v \<in> V" shows "((\<sharp>\<cdot>) ((1::'f) / ordG) \<circ> (\<Sum>g\<in>G. (*\<cdot>) (- g) \<circ> (\<lambda>a. if a \<in> (\<Oplus>G\<leftarrow>[W, U]. G) then W\<oplus>U\<leftarrow> a ! 1 else (0::'v)) \<circ> (*\<cdot>) g)) (((\<sharp>\<cdot>) ((1::'f) / ordG) \<circ> (\<Sum>g\<in>G. (*\<cdot>) (- g) \<circ> (\<lambda>a. if a \<in> (\<Oplus>G\<leftarrow>[W, U]. G) then W\<oplus>U\<leftarrow> a ! 1 else (0::'v)) \<circ> (*\<cdot>) g)) v) = ((\<sharp>\<cdot>) ((1::'f) / ordG) \<circ> (\<Sum>g\<in>G. (*\<cdot>) (- g) \<circ> (\<lambda>a. if a \<in> (\<Oplus>G\<leftarrow>[W, U]. G) then W\<oplus>U\<leftarrow> a ! 1 else (0::'v)) \<circ> (*\<cdot>) g)) v"
Mapping_lookup_upd_keys_on: fixes m :: "('b, 'a) mapping"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and X :: "'b set"   and a :: "'b" shows "Mapping.lookup (upd_keys_on m f X) a = (case Mapping.lookup m a of None \<Rightarrow> None | Some b \<Rightarrow> Some (if a \<in> X then f a b else b))"
Says_kas_message_form: fixes A :: "agent"   and authK :: "nat"   and Ta :: "msg"   and authTicket :: "msg"   and evs :: "event list" assumes "Gets A (Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Ta, authTicket\<rbrace>) \<in> set evs"   and "evs \<in> kerbIV_gets" shows "authK \<notin> range shrK \<and> authK \<in> symKeys \<and> authTicket = Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Ta\<rbrace> \<or> authTicket \<in> analz (knows Spy evs)"
pareto_loser_in_alts: fixes x :: "'alt pmf" assumes "x \<in> pareto_losers (SD \<circ> R)" shows "x \<in> lotteries_on alts"
shift_closed: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "f \<in> carrier P" shows "shift n f \<in> carrier P"
Un_empty: fixes A :: "'e set"   and B :: "'e set" shows "(A \<union> B = {}) = (A = {} \<and> B = {})"
solves_ode_congI: fixes x :: "real \<Rightarrow> 'd"   and f :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "real set"   and X :: "'d set"   and y :: "real \<Rightarrow> 'd"   and g :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and S :: "real set"   and Y :: "'d set" assumes "(x solves_ode f) T X"   and "\<And>t. t \<in> T \<Longrightarrow> x t = y t"   and "\<And>t. t \<in> T \<Longrightarrow> f t (x t) = g t (x t)"   and "T = S"   and "X = Y" shows "(y solves_ode g) S Y"
coinduct: fixes X :: "enat \<Rightarrow> enat \<Rightarrow> bool"   and x1 :: "enat"   and x2 :: "enat" assumes "X x1 x2"   and "\<And>x1 x2. X x1 x2 \<Longrightarrow> (\<exists>n. x1 = 0 \<and> x2 = n) \<or> (\<exists>m n k. x1 = eSuc m \<and> x2 = n + k \<and> (X m n \<or> (m, n) \<in> Le_enat) \<and> k \<noteq> 0)" shows "(x1, x2) \<in> Le_enat"
set_break_down_left: fixes M :: "'a multiset"   and N :: "'a set multiset" shows "{#m \<in># M \<times># N. fst m \<in> snd m#} = (\<Sum>m\<in>#M. {#m#} \<times># filter_mset ((\<in>) m) N)"
g: shows "cmod (g a) \<le> pi + cmod (f a)"
open_outside: fixes S :: "'a set" assumes "closed S" shows "open (outside S)"
CardinalityHeapSafe_axioms: shows "CardinalityHeapSafe cHeap Aheap"
union_disjoint: fixes A :: "'a set"   and B :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "finite A"   and "finite B"   and "A \<inter> B = {}" shows "prod g (A \<union> B) = prod g A * prod g B"
succ_smallest: fixes i :: "'a"   and j :: "'a" assumes "(i, j) \<in> r"   and "i \<noteq> j" shows "(succ i, j) \<in> r"
Ex_E: fixes A :: "fm"   and H :: "fm set"   and B :: "fm"   and i :: "name" assumes "insert A H \<turnstile> B"   and "atom i \<sharp> B"   and "\<forall>C\<in>H. atom i \<sharp> C" shows "insert (SyntaxN.Ex i A) H \<turnstile> B"
empty_rep: fixes q :: "('c, 'd) SkewBinomialHeap" shows "(q = local.empty) = (Rep_SkewBinomialHeap q = BsSkewBinomialHeapStruc.bs_empty)"
associated_rel_def: shows "associated_rel = {(a, b). normalize a = normalize b}"
orderE: fixes a :: "'g"   and b :: "'g" assumes "a \<le> b" obtains "a = min a b"
mcont2: fixes out :: "'b" shows "mcont resumption.lub_fun resumption.le_fun lub ord (\<lambda>c. g out c (Resumption.resumption.Pause out c))"
MP: fixes M :: "('fun, 'var) Term.term \<Rightarrow> bool"   and t :: "('fun, 'var) Term.term" assumes "M t" shows "SMPp M t"
assms_8: fixes tx :: "real \<times> 'a" assumes "tx \<in> T \<times> X" shows "(f' has_derivative g'' tx) (at tx)"
DirProd_group: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme" assumes "Group.group G"   and "Group.group H" shows "Group.group (G \<times>\<times> H)"
le_Inf_iff: fixes A :: "'a set"   and b :: "'a" shows "(\<Squnion> A \<le> b) = (\<forall>a\<in>A. a \<le> b)"
wf_2: shows "prefs_from_table_wf agents alts ys"
af_congruent_axioms: shows "af_congruent (\<sim>\<^sub>P)"
has_free_generator_nonzeroring: fixes H :: "'a set" assumes "free_generator R M H"   and "\<exists>p\<in>linear_span R M (carrier R) H. p \<noteq> \<zero>" shows "\<not> zeroring R"
nabla_unfold_eq: fixes x :: "('a \<times> 'a) set" shows "relfdia x (rel_nabla x) = rel_nabla x"
lincomb_list_alt2: fixes s :: "'a vec list"   and ls :: "nat list"   and c :: "nat \<Rightarrow> 'a" assumes "\<And>v. v \<in> set s \<Longrightarrow> dim_vec v = n"   and "\<And>i. i \<in> set ls \<Longrightarrow> i < length s" shows "M.sumlist (map2 (\<lambda>x y. x \<cdot>\<^sub>v s ! y) (map c ls) ls) = M.sumlist (map2 (\<lambda>x y. x \<cdot>\<^sub>v s ! y) (map c (filter (\<lambda>i. c i \<noteq> (0::'a)) ls)) (filter (\<lambda>i. c i \<noteq> (0::'a)) ls))"
fold_closed_eq: fixes A :: "'a set"   and B :: "'b set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b" assumes "\<And>a b. \<lbrakk>a \<in> A; b \<in> B\<rbrakk> \<Longrightarrow> f a b = g a b"   and "\<And>a b. \<lbrakk>a \<in> A; b \<in> B\<rbrakk> \<Longrightarrow> g a b \<in> B"   and "z \<in> B" shows "Finite_Set.fold f z A = Finite_Set.fold g z A"
run_sim2: fixes r :: "nat \<Rightarrow> 'v" assumes "G'.is_run r" shows "is_run (fi \<circ> r)"
ordLeq_ordLess_trans: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set"   and r'' :: "('f \<times> 'f) set" assumes "r \<le>o r'"   and "r' <o r''" shows "r <o r''"
vars_monom_subset: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'a" shows "vars (MPoly_Type.monom m a) \<subseteq> keys m"
eq5: shows "(\<Sum>i = Suc j..card X. int i - int (aa i) - int j - 1 gchoose (i - j) + (int i - int (bb i) - int j - 1 gchoose (i - j))) = (\<Sum>i = Suc j..card X. (- 1) ^ (i - j) * (int (aa i) gchoose (i - j) + (int (bb i) gchoose (i - j))))"
elims: fixes x :: "'c dres"   and xa :: "'c \<Rightarrow> 'b dres"   and y :: "'b dres" assumes "x \<bind> xa = y" obtains uu_ :: "'c \<Rightarrow> 'b dres" where "x = dFAIL" and "xa = uu_" and "y = dFAIL"   | uv_ :: "'c \<Rightarrow> 'b dres" where "x = dSUCCEED" and "xa = uv_" and "y = dSUCCEED"   | xb :: "'c" and f :: "'c \<Rightarrow> 'b dres" where "x = dRETURN xb" and "xa = f" and "y = f xb"
injective_not_constant: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "open S"   and "inj_on f S"   and "f constant_on S" shows "S = {}"
preserved: fixes M :: "moebius"   and u :: "complex_homo"   and v :: "complex_homo" assumes "unit_disc_fix M"   and "u \<in> unit_disc"   and "v \<in> unit_disc"   and "P (moebius_pt M u) (moebius_pt M v)" shows "P u v"
RCOSETS_def: fixes G :: "('a, 'b) monoid_scheme"   and H :: "'a set" shows "rcosets\<^bsub>G\<^esub> H = (\<Union>a\<in>carrier G. {H #>\<^bsub>G\<^esub> a})"
is_proper_suffix_def: fixes u :: "'a list"   and v :: "'a list" shows "is_proper_suffix u v = (\<exists>w. w \<noteq> [] \<and> w @ u = v)"
HPhase1or2ReadThen_HInv1: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "HInv1 s"   and "HPhase1or2ReadThen s s' p d q" shows "HInv1 s'"
product_topology_basis: fixes T :: "'c \<Rightarrow> 'd topology"   and X :: "'c \<Rightarrow> 'd set"   and I :: "'c set" assumes "\<And>i. openin (T i) (X i)"   and "finite {i. X i \<noteq> topspace (T i)}" shows "openin (product_topology T I) (Pi\<^sub>E I X)"
strict_coboundedI2: fixes c :: "'a Word.word"   and b :: "'a Word.word"   and a :: "'a Word.word" assumes "c <s b" shows "c <s signed.max a b"
ih': fixes s :: "('s, 'v) tm" assumes "s \<in> set ss" shows "eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt s)) = eval_tpoly B (map_tpoly (\<lambda>x. x) zhmset_of (wt s))"
lbisimulation_refl: shows "lbisimulation Id pid_system_automaton pid_system_automaton"
less_le_mult': fixes w :: "int"   and c :: "int"   and b :: "int" assumes "w * c < b * c"   and "0 \<le> c" shows "(w + 1) * c \<le> b * c"
sel_2: fixes x21 :: "nat"   and x22 :: "nat" shows "phil (putsdown x21 x22) = x21"
sqrt_Suc: fixes n :: "nat" shows "Discrete.sqrt (Suc n) = (if \<exists>m. Suc n = m\<^sup>2 then Suc (Discrete.sqrt n) else Discrete.sqrt n)"
lift2_def: fixes h :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and inp' :: "'a \<Rightarrow> 'c option"   and inp :: "'a \<Rightarrow> 'd option" shows "lift2 h inp' inp \<equiv> \<lambda>i. case (inp' i, inp i) of (None, b) \<Rightarrow> None | (Some v', None) \<Rightarrow> None | (Some v', Some v) \<Rightarrow> Some (h v' v)"
lookup_pair_eq_valueI: fixes xs :: "('a \<times> 'c) list"   and k :: "'a"   and v :: "'c" assumes "cmp2.oalist_inv_raw xs"   and "(k, v) \<in> set xs" shows "cmp2.lookup_pair xs k = v"
hyps_3: shows "\<alpha> (kill_short (G -- choose_v G k) k) \<le> \<alpha> (G -- choose_v G k)"
A1: fixes m :: "nat" shows "s m \<in> carrier Zp"
more_than_one_mset_mset_diff: fixes a :: "'a"   and M :: "'a multiset" assumes "a \<in># remove1_mset a M" shows "set_mset (remove1_mset a M) = set_mset M"
partition_axioms: shows "Set_Theory.partition (Sigma I \<FF>2) e2.Partition"
arcs_ends_mono: fixes H :: "('a, 'b) pre_digraph"   and G :: "('a, 'b) pre_digraph" assumes "subgraph H G" shows "arcs_ends H \<subseteq> arcs_ends G"
FreeGroup_0_closed: fixes S :: "'b set" shows "0 \<in> FreeGroup S"
simps_5: fixes r :: "real"   and x :: "octo" shows "Im4 (r *\<^sub>R x) = r * Im4 x"
if_false_eval\<^sub>w: fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and mem :: "'a \<Rightarrow> 'b"   and b :: "'d"   and t :: "('a, 'c, 'd) Stmt"   and c' :: "('a, 'c, 'd) Stmt"   and mds :: "Mode \<Rightarrow> 'a set" assumes "sifum_lang_no_dma eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars"   and "\<not> eval\<^sub>B mem b" shows "(((Stmt.If b t c', mds), mem), (c', mds), mem) \<in> sifum_lang_no_dma.eval\<^sub>w eval\<^sub>A eval\<^sub>B"
IH: fixes y :: "nat"   and xs :: "'a list"   and t :: "'a tree"   and zs :: "'a list" assumes "y < n"   and "y \<le> length xs"   and "bal y xs = (t, zs)" shows "height t = nat \<lceil>log 2 (real (y + 1))\<rceil>"
\<tau>diverge_bisim_inv: fixes s1 :: "'a"   and s2 :: "'c" assumes "bisim12 s1 s2" shows "s1 -\<tau>1\<rightarrow> \<infinity> = s2 -\<tau>2\<rightarrow> \<infinity>"
ts: fixes x\<^sub>0 :: "nat"   and x\<^sub>1 :: "nat"   and k :: "nat"   and as :: "real list"   and bs :: "real list"   and ts :: "(nat \<Rightarrow> nat) list"   and i :: "nat" assumes "akra_bazzi_recursion x\<^sub>0 x\<^sub>1 k as bs ts"   and "i < length bs" shows "akra_bazzi_term x\<^sub>0 x\<^sub>1 (bs ! i) (ts ! i)"
take_all: fixes xs :: "'a list"   and n :: "nat" assumes "|xs| \<le> n" shows "take n xs = xs"
sub_unit_closed: shows "\<one> \<in> R\<setminus>I"
idem: fixes a :: "'a" shows "a \<^bold>\<sqinter> a = a"
simplex_map: fixes x :: "'a set" assumes "x \<in> X" shows "f ` x \<in> Y"
insert_5: shows "\<forall>s\<in>insert x f. \<exists>p. p division_of s"
of_int_eq: fixes n :: "int"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a"   and z :: "int" assumes "mod_ring n Rep Abs" shows "of_int z = Abs (z mod n)"
ns: fixes r :: "rat"   and a :: "QDelta atom" assumes "(r, a) \<in> set C" shows "ns (r, a) \<in> snd ` set cs \<and> is_leq_ns (s (r, a) *R ns (r, a)) \<and> s (r, a) \<noteq> 0 \<and> subst_poly_lec \<sigma> (lec_of_nsc (r *R nsc_of_atom a)) = lec_of_nsc (s (r, a) *R ns (r, a))"
witness_submatrix: shows "submatrix Aw' rows_with_1 rows_with_1 = 1\<^sub>m (r ^ N_half)"
stern_brocot_unfold: shows "stern_brocot_recurse = Node (1, 1) (pure recip \<diamondop> (pure succ \<diamondop> (pure recip \<diamondop> stern_brocot_recurse))) (pure succ \<diamondop> stern_brocot_recurse)"
ApproximationTr1: fixes aa :: "'a"   and N :: "nat" assumes "aa \<in> carrier K" shows "1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a aa^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K"
large_n: shows "\<forall>n\<ge>N. B * b n = c n * a n - c (n + 1) \<and> real_of_int \<bar>c (n + 1)\<bar> < real_of_int (a n) / 2 \<and> 2 \<le> a n"
Vfrom_mono2: fixes a :: "V"   and i :: "V"   and j :: "V" shows "Vfrom a i \<subseteq>\<^sub>\<circ> Vfrom a (i \<union>\<^sub>\<circ> j)"
scale_left_imp_eq: fixes a :: "rat"   and x :: "'a"   and y :: "'a" assumes "a \<noteq> 0"   and "a *R x = a *R y" shows "x = y"
rel_generat_reflI: fixes x :: "('a, 'b, 'c) generat"   and A :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and C :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>y. y \<in> generat_pures x \<Longrightarrow> A y y"   and "\<And>out. out \<in> generat_outs x \<Longrightarrow> B out out"   and "\<And>cont. cont \<in> generat_conts x \<Longrightarrow> C cont cont" shows "rel_generat A B C x x"
apply_adaptation_def: fixes first :: "bool"   and mem :: "'Var \<Rightarrow> 'Val"   and A :: "'Var \<Rightarrow> ('Val \<times> 'Val) option" shows "apply_adaptation first mem A = (\<lambda>x. case A x of None \<Rightarrow> mem x | Some (v\<^sub>1, v\<^sub>2) \<Rightarrow> if first then v\<^sub>1 else v\<^sub>2)"
InvariantsAfterExhaustiveUnitPropagate: fixes state :: "State"   and F0 :: "Literal list list"   and Vbl :: "nat set" assumes "exhaustiveUnitPropagate_dom state"   and "InvariantConsistent (getM state)"   and "InvariantUniq (getM state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)"   and "InvariantConflictFlagCharacterization (getConflictFlag state) (getF state) (getM state)"   and "InvariantQCharacterization (getConflictFlag state) (getQ state) (getF state) (getM state)"   and "InvariantUniqQ (getQ state)"   and "InvariantVarsQ (getQ state) F0 Vbl"   and "InvariantVarsM (getM state) F0 Vbl"   and "InvariantVarsF (getF state) F0 Vbl" shows "let state' = exhaustiveUnitPropagate state in InvariantConsistent (getM state') \<and> InvariantUniq (getM state') \<and> InvariantWatchListsContainOnlyClausesFromF (getWatchList state') (getF state') \<and> InvariantWatchListsUniq (getWatchList state') \<and> InvariantWatchListsCharacterization (getWatchList state') (getWatch1 state') (getWatch2 state') \<and> InvariantWatchesEl (getF state') (getWatch1 state') (getWatch2 state') \<and> InvariantWatchesDiffer (getF state') (getWatch1 state') (getWatch2 state') \<and> InvariantWatchCharacterization (getF state') (getWatch1 state') (getWatch2 state') (getM state') \<and> InvariantConflictFlagCharacterization (getConflictFlag state') (getF state') (getM state') \<and> InvariantQCharacterization (getConflictFlag state') (getQ state') (getF state') (getM state') \<and> InvariantUniqQ (getQ state') \<and> InvariantVarsQ (getQ state') F0 Vbl \<and> InvariantVarsM (getM state') F0 Vbl \<and> InvariantVarsF (getF state') F0 Vbl"
UN: fixes K :: "'c set set"   and S :: "'c set"   and B :: "'c set set" assumes "\<And>U. U \<in> K \<Longrightarrow> generated_topology S B U" shows "generated_topology S B (\<Union> K)"
cases: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and a :: "'a stream" assumes "alw \<phi> a" obtains xs :: "'a stream" where "a = xs" and "\<phi> xs" and "alw \<phi> (stl xs)"
keys_tail_less_lt: fixes v :: "'t"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "v \<in> keys (tail p)" shows "v \<prec>\<^sub>t lt p"
rel_conversep: fixes R :: "'d \<Rightarrow> 'c \<Rightarrow> bool" shows "rel_option R\<inverse>\<inverse> = (rel_option R)\<inverse>\<inverse>"