hcju/isabelleps huggingface dataset:
to_Gromov_completion_lim: fixes u :: "'c \<Rightarrow> 'd"   and a :: "'d"   and F :: "'c filter" shows "((\<lambda>n. to_Gromov_completion (u n)) \<longlongrightarrow> to_Gromov_completion a) F = (u \<longlongrightarrow> a) F"
prod_ereal_pos: fixes I :: "'a set"   and f :: "'a \<Rightarrow> ereal" assumes "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i" shows "0 \<le> prod f I"
atomize_vectorize_poly: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" shows "atomize_poly (vectorize_poly p) = p"
prv_imp_triv: fixes \<phi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<psi> \<in> fmla"   and "prv \<psi>" shows "prv (imp \<phi> \<psi>)"
normalize_idem: fixes a :: "'b"   and b :: "'b" shows "normalize (gcd a b) = gcd a b"
verit_sko_ex_indirect: fixes x :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "x = (SOME x. P x)" shows "(\<exists>x. P x) = P x"
aprimedivisor_dvd': fixes n :: "'a" assumes "n \<noteq> (0::'a)"   and "\<not> is_unit n" shows "aprimedivisor n dvd n"
mf: shows "monotone sum_bot.le_fun sum_bot_ord B"
export_s: fixes G :: "'a triple set"   and P :: "'a assn"   and c :: "com"   and Q :: "'a assn" assumes "\<And>s'. G|-{\<lambda>Z s. s' = s \<and> P Z s}. c .{Q}" shows "G|-{P}. c .{Q}"
set_intros_2: fixes y :: "'d"   and x22 :: "'d list"   and x21 :: "'d" assumes "y mem x22" shows "y mem (x21 # x22)"
HPhase1or2ReadThen_HInv4a1_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "HPhase1or2ReadThen s s' p d q"   and "HInv4a1 s p" shows "HInv4a1 s' p"
hfunction_hunion: fixes f :: "hf"   and g :: "hf" assumes "hdomain f \<sqinter> hdomain g = 0" shows "hfunction (f \<squnion> g) = (hfunction f \<and> hfunction g)"
LeftDerives1_def: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and u :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and v :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>" shows "CFG.LeftDerives1 \<NN> \<TT> \<RR> u i r v = (CFG.leftmost \<NN> \<TT> i u \<and> CFG.Derives1 \<NN> \<TT> \<RR> u i r v)"
sSum_unique: fixes N :: "('g, 'b, 'h) Module_scheme"   and H :: "'a set"   and H1 :: "'a set"   and h :: "'a"   and x1 :: "'a"   and x2 :: "'a"   and y1 :: "'a"   and y2 :: "'a" assumes "R module N"   and "free_generator R M H"   and "H1 \<subseteq> H"   and "h \<in> H - H1"   and "x1 \<in> fgs R M H1"   and "x2 \<in> fgs R M H1"   and "y1 \<in> fgs R M {h}"   and "y2 \<in> fgs R M {h}"   and "x1 \<plusminus> y1 = x2 \<plusminus> y2" shows "x1 = x2 \<and> y1 = y2"
lbound_ofD2: fixes x :: "'a"   and y :: "'a"   and b :: "'a" assumes "dual_order.lbound_of x y b" shows "order.greater_eq b y"
Arr_iff_dom_in_domain: fixes f :: "'a" shows "right (src \<mu>) f = (local.dom f \<in> Right.domains f)"
intro: fixes L :: "('c, 'd) gorder_scheme" assumes "upper_semilattice L"   and "lower_semilattice L" shows "lattice L"
IQ_in: fixes l :: "char list \<Rightarrow> nat"   and s :: "char list \<Rightarrow> int" assumes "I l s"   and "bval b s" shows "S s = S (postQ C s)"
interval_lebesgue_integral_cong_AE: fixes f :: "real \<Rightarrow> 'a"   and M :: "real measure"   and g :: "real \<Rightarrow> 'a"   and a :: "ereal"   and b :: "ereal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "a \<le> b"   and "AE x\<in>einterval a b in M. f x = g x"   and "einterval a b \<in> sets M" shows "interval_lebesgue_integral M a b f = interval_lebesgue_integral M a b g"
simps_15: fixes f :: "'i \<Rightarrow> 'h" shows "sum_bot_map f Bottom = Bottom"
lexordp_conv_lexord: fixes xs :: "char list"   and ys :: "char list" shows "char.lexordp xs ys = ((xs, ys) \<in> lexord {(x, y). less_char x y})"
finite_lin_indpt2: fixes A :: "'c set" assumes "finite A"   and "A \<subseteq> carrier V"   and "\<And>a. \<lbrakk>a \<in> A \<rightarrow> carrier K; lincomb a A = \<zero>\<^bsub>V\<^esub>\<rbrakk> \<Longrightarrow> \<forall>v\<in>A. a v = \<zero>\<^bsub>K\<^esub>" shows "lin_indpt A"
image_gcsrp: fixes H :: "'a set" assumes "G \<guillemotright> H " shows "gcsrp G H (gcsrp_map G H ` set_rcs G H)"
but_last_zip: fixes xs :: "'b list"   and ys :: "'c list" assumes "length xs = length ys"   and "xs \<noteq> []" shows "last (zip xs ys) = (last xs, last ys) \<and> butlast (zip xs ys) = zip (butlast xs) (butlast ys)"
filter_filter_mset: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool"   and M :: "'c multiset" shows "filter_mset P (filter_mset Q M) = {#x \<in># M. Q x \<and> P x#}"
path_connected_frontier: fixes S :: "'b set"   and T :: "'b set" assumes "path_connected S"   and "S \<inter> T \<noteq> {}"   and "S \<inter> - T \<noteq> {}" obtains s :: "'b" where "s \<in> S" and "s \<in> frontier T"
is_regular_has_right_codom: fixes U :: "'a set set"   and \<pp> :: "'a set"   and s :: "'a set \<Rightarrow> ('a \<times> 'a) set" assumes "U \<subseteq> Spec"   and "\<pp> \<in> U"   and "is_regular s U" shows "s \<pp> \<in> R\<setminus>\<pp> \<^sup>\<inverse> R \<^bsub>(+) (\<cdot>) \<zero>\<^esub>"
isomorphic_relative_homology_groups_inclusion_contractible: fixes X :: "'a topology"   and S :: "'a set"   and T :: "'a set"   and p :: "int" assumes "contractible_space X"   and "contractible_space (subtopology X S)"   and "T \<subseteq> S"   and "topspace X \<inter> S \<noteq> {}" shows "relative_homology_group p (subtopology X S) T \<cong> relative_homology_group p X T"
pochhammer_0: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b" assumes "class.comm_semiring_1 times one plus zero" shows "comm_semiring_1.pochhammer times one plus zero a 0 = one"
finfun_default_const: fixes b :: "'a" shows "finfun_default (finfun_const b) = (if finite UNIV then undefined else b)"
gstep_sim1: fixes c :: "('c, 'ls, 'gs) global_config"   and c' :: "('c, 'ls, 'gs) global_config"   and d :: "('d, 'ls, 'gs) global_config" assumes "(c, c') \<in> s1.gstep"   and "rel_gc c d" shows "\<exists>d'. rel_gc c' d' \<and> (d, d') \<in> s2.gstep"
simps_1: fixes dt :: "loc \<Rightarrow> ty option" shows "typeof dt Unit = Some (PrimT Void)"
assms_1: assumes "\<phi> \<pi>l" shows "\<psi> \<pi>l"
natural_transformation_axioms: shows "natural_transformation C C Adj.FG.map Adj.FG.map Adj.FG.map"
assms_3: shows "splits_at (ladder_\<alpha> \<alpha> D L 1) (ladder_i L 1) a1 A a2"
ball_min_Int: fixes a :: "'a"   and r :: "real"   and s :: "real" shows "ball a (min r s) = ball a r \<inter> ball a s"
existence_ivl_initial_time_iff: fixes t0 :: "real"   and x0 :: "'a" shows "(t0 - t0 \<in> rev.existence_ivl0 x0) = (t0 \<in> UNIV \<and> x0 \<in> X)"
poly_shift_cfs: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "f \<in> carrier P" shows "poly_shift f n = f (Suc n)"
inj_on_inverseI: fixes A :: "'b set"   and g :: "'d \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'd" assumes "\<And>x. x \<in> A \<Longrightarrow> g (f x) = x" shows "inj_on f A"
last_i_take2: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "0 < n" shows "last (f \<Down> n) = f (n - Suc 0)"
i_Exec_Stream_Init_eq_i_Exec_Stream_Cons_output: fixes output_fun :: "'e \<Rightarrow> 'd message_af"   and c :: "'e"   and trans_fun :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and input :: "nat \<Rightarrow> 'f" assumes "output_fun c = NoMsg" shows "output_fun \<circ> i_Exec_Comp_Stream_Init trans_fun input c = [NoMsg] \<frown> (output_fun \<circ> i_Exec_Comp_Stream trans_fun input c)"
has_subprob_density_measurable: fixes M :: "'b measure"   and N :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" assumes "has_subprob_density M N f" shows "f \<in> borel_measurable N"
comm: fixes a :: "'d DBMEntry"   and b :: "'d DBMEntry" shows "dbm_add a b = dbm_add b a"
p_pow_nonzero': fixes n :: "nat" shows "[p] \<cdot> \<one> [^] n \<in> nonzero Zp"
Node_2: fixes ps :: "(real, 'a) vec list" assumes "set_kdt r \<union> set ps - set (nearest_nbors n ps p r) \<noteq> {}" shows "length (nearest_nbors n ps p r) = n"
eucl_truncate_up_zero: fixes p :: "nat" shows "eucl_truncate_up p (0::'a) = (0::'a)"
F_n_integral: fixes f :: "bool stream \<Rightarrow> real"   and n :: "nat" assumes "f \<in> borel_measurable (F n)" shows "has_bochner_integral M f (\<Sum>w\<in>range (pseudo_proj_True n). prob (pseudo_proj_True n -` {w} \<inter> space M) * f w)"
norm_divide: fixes a :: "'a"   and b :: "'a" shows "norm (a / b) = norm a / norm b"
memo_lookup: fixes f :: "'v \<Rightarrow> 'm"   and x :: "'v"   and g :: "int \<Rightarrow> int \<Rightarrow> 'm" shows "bind (memo f x) (\<lambda>i. bind (lookup x) (g i)) = bind (memo f x) (\<lambda>i. g i i)"
return_partition_basics_1: fixes A :: "'a set"   and n :: "nat" assumes "A \<in> sets M" shows "return_partition A n \<in> sets M"
Aheap_heap3: fixes x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and a :: "Arity" assumes "x \<in> thunks \<Gamma>"   and "many \<sqsubseteq> (cHeap \<Gamma> e\<cdot>a) x" shows "(Aheap \<Gamma> e\<cdot>a) x = up\<cdot>0"
atLeastatMost_subset_iff: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset"   and d :: "'b multiset" shows "(subset_mset.atLeastAtMost a b \<subseteq> subset_mset.atLeastAtMost c d) = (\<not> a \<subseteq># b \<or> c \<subseteq># a \<and> b \<subseteq># d)"
subterms_subset: fixes t' :: "('a, 'b) Term.term"   and t :: "('a, 'b) Term.term" assumes "t' \<sqsubseteq> t" shows "subterms t' \<subseteq> subterms t"
finite_Un: fixes F :: "'a set"   and G :: "'a set" shows "finite (F \<union> G) = (finite F \<and> finite G)"
left_neutral: fixes a :: "'b upred" shows "(false \<or> a) = a"
in_rel_Gr: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "in_rel (BNF_Def.Gr A f) = BNF_Def.Grp A f"
simps_6: fixes x2 :: "('b, 'c, 'd, 'e) Tree"   and x1 :: "('b, 'c, 'd, 'e) Tree set['b]" shows "tNot x2 \<noteq> tConj x1"
oinvariantD: fixes A :: "('a \<times> 'b, 'c) automaton"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and U :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<times> 'b \<Rightarrow> bool"   and s :: "'a \<times> 'b" assumes "A \<Turnstile> (S, U \<rightarrow>) P"   and "s \<in> oreachable A S U" shows "P s"
seq_compact_imp_Heine_Borel: fixes S :: "'b set" assumes "seq_compact S" shows "compact S"
assms_2: shows "bob \<le> vimage2p rhb rhb (=)"
remDupsNMT: fixes p :: "'b list" assumes "p \<noteq> []" shows "remdups p \<noteq> []"
F_partially_differentiable: shows "\<forall>a\<in>Dx_pair. has_partial_vector_derivative (\<lambda>x. F x \<bullet> j) i (partial_vector_derivative (\<lambda>a. F a \<bullet> j) i a) a"
integrable_const_bound: fixes f :: "real \<Rightarrow> 'c"   and B :: "real"   and sa :: "'a" assumes "AE x in exponential (escape_rate sa). norm (f x) \<le> B"   and "E.random_variable sa borel f" shows "integrable (exponential (escape_rate sa)) f"
distinct_240: fixes x13 :: "Syntax.term"   and x101 :: "Syntax.term"   and x102 :: "Syntax.term"   and x103 :: "Syntax.term" shows "Roll x13 \<noteq> Case x101 x102 x103"
power_dvd_iff_le_multiplicity: fixes n :: "nat" shows "(p ^ n dvd x) = (n \<le> multiplicity p x)"
mult_poly_add_left: fixes p :: "'b poly"   and q :: "'b poly"   and r :: "'b poly" shows "(p + q) * r = p * r + q * r"
assms: shows "\<Gamma> \<turnstile> e, eP, Roll eV' : Mu \<alpha> \<tau>"
rel_stacktraces_Cons: fixes get :: "'c \<Rightarrow> ('d, 'e, 'c, 'op, 'opinl, 'opubx, 'f, 'g) Ubx.instr fundef option"   and st1 :: "('c, 'dyn) frame list"   and st2 :: "('c, ('dyn, 'ubx1, 'ubx2) unboxed) frame list"   and f :: "'c"   and \<Sigma>1 :: "'dyn list"   and \<Sigma>2 :: "('dyn, 'ubx1, 'ubx2) unboxed list"   and fd2 :: "('d, 'e, 'c, 'op, 'opinl, 'opubx, 'f, 'g) Ubx.instr fundef"   and n :: "nat"   and opt :: "'c option" assumes "rel_stacktraces get st1 st2 (Some f)"   and "\<Sigma>1 = norm_stack \<Sigma>2"   and "get f = Some fd2"   and "sp_fundef get fd2 (take n (body fd2)) = Ok (map typeof \<Sigma>2)"   and "pred_option (is_valid_fun_call get fd2 n \<Sigma>2) opt" shows "rel_stacktraces get (Frame f n \<Sigma>1 # st1) (Frame f n \<Sigma>2 # st2) opt"
assignsE_good_approx: fixes Env :: "'a env_scheme"   and s0 :: "state"   and e :: "expr"   and v :: "val"   and s1 :: "state" assumes "prg Env\<turnstile>s0 \<midarrow>e-\<succ>v\<rightarrow> s1"   and "normal s1" shows "assignsE e \<subseteq> dom (locals (snd s1))"
sum_FS'_B: fixes i :: "'a" shows "(\<Sum>x\<in>F \<inter> S' i. norm (f x)) \<le> B i"
ranI: fixes m :: "'e \<Rightarrow> 'd option"   and a :: "'e"   and b :: "'d" assumes "m a = Some b" shows "b \<in> ran m"
indexed_eval_is_weak_ring_morphism: fixes q :: "('c multiset \<Rightarrow> 'a) list" assumes "q \<in> carrier (poly_ring L)" shows "weak_ring_morphism (eval_pmod q) (PIdl\<^bsub>poly_ring L\<^esub> q) (poly_ring L)"
induced_automorphism_fg_symmetric: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "OpposedThinChamberComplexFoldings.induced_automorphism X f g = OpposedThinChamberComplexFoldings.induced_automorphism X g f"
decomposeLS_fragmentsLS: fixes c :: "('g, 'h, 'i, 'j) com"   and ictxt :: "('g, 'h, 'i, 'j) com \<Rightarrow> ('g, 'h, 'i, 'j) com"   and fctxt :: "('g, 'h, 'i, 'j) com \<Rightarrow> ('g, 'h, 'i, 'j) com list"   and s :: "('g, 'h, 'i, 'j) com list \<times> 'h option \<times> 'j" assumes "(c, ictxt, fctxt) \<in> decomposeLS s" shows "(c, loc_compC c (fctxt c @ tl (cPGM s))) \<in> fragmentsLS s"
assms_1: shows "k \<le> 2 ^ n - 1"
ln_o_ln: shows "(\<lambda>x. ln (f x)) \<in> o(\<lambda>x. ln (g x))"
tighten_poly_bounds_def: fixes p :: "int poly"   and l :: "rat"   and r :: "rat"   and sr :: "rat" shows "tighten_poly_bounds p l r sr = (let m = (l + r) / 2; sm = sgn (ipoly p m) in if sm = sr then (l, m, sm) else (m, r, sr))"
prems: shows "list_all tfr\<^sub>s\<^sub>t\<^sub>p (x # S)"
ipv4addr_of_dotdecimal_bit: fixes a :: "nat"   and b :: "nat"   and c :: "nat"   and d :: "nat" shows "ipv4addr_of_dotdecimal (a, b, c, d) = (ipv4addr_of_nat a << 24) + (ipv4addr_of_nat b << 16) + (ipv4addr_of_nat c << 8) + ipv4addr_of_nat d"
unat_eq_nat_uint: fixes w :: "'c word" shows "unat w = nat (uint w)"
Beta_fresh: fixes s :: "Lambda_Z.term"   and t :: "Lambda_Z.term"   and x :: "name" assumes "s \<rightarrow>\<^sub>\<beta> t"   and "atom x \<sharp> s" shows "atom x \<sharp> t"
assms: shows "\<forall>x\<in>set xs. \<not> gt x x"
fls_base_factor_base: fixes f :: "'a fls" shows "fls_nth (fls_base_factor f) (fls_subdegree (fls_base_factor f)) = fls_nth f (fls_subdegree f)"
observable_moves_equal_length_1: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as :: "'edge list"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "S,f \<turnstile> (ms,s) =as\<Rightarrow> (ms',s')" shows "length ms = length s"
save_restore_instr_result: fixes i :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "fst i \<in> {ctrl_type SAVE, ctrl_type RESTORE}" shows "snd (save_restore_instr i s) = False"
poly_mapping_size_def: fixes f :: "'b \<Rightarrow> nat"   and g :: "'c \<Rightarrow> nat"   and m :: "'b \<Rightarrow>\<^sub>0 'c" shows "poly_mapping_size f g m = g (0::'c) + (\<Sum>k\<in>keys m. Suc (f k + g (lookup m k)))"
decrpoly_zero: fixes p :: "'b poly" shows "(decrpoly p = 0\<^sub>p) = (p = 0\<^sub>p)"
B: shows "{c a} \<inter> range c \<inter> C = {c a}"
add_order_eq0: fixes a :: "'a"   and n :: "nat" assumes "add_order a = 0"   and "order.greater n 0" shows "a +^ n \<noteq> (0::'a)"
I_minWeightBasisD_4: fixes T :: "'a set"   and E :: "'a set"   and x :: "'a"   and y :: "'a" assumes "I_minWeightBasis (T, E)"   and "x \<in> T"   and "y \<in> E" shows "weight x \<le> weight y"
primroot_take: fixes u :: "'b list" assumes "u \<noteq> \<epsilon>" shows "\<rho> u = take \<^bold>|\<rho> u\<^bold>| u"
f_2: shows "(f has_contour_integral j) (linepath c b)"
cong2_per2__cong: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p" assumes "Per A B C"   and "Per A' B' C'"   and "Cong A C A' C'"   and "Cong B C B' C'" shows "Cong B A B' A'"
assms: shows "\<forall>qd1 qd2. (qd1, qd2) \<in> set qd \<longrightarrow> qd1 \<le> qd2"
simps: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" shows "mat_map f (a, b, c, d) = (f a, f b, f c, f d)"
chamber_system_image: shows "f \<turnstile> \<C> = \<C> \<inter> (f \<turnstile> X)"
SourcesA11_L1: shows "Sources level1 sA11 = {}"
ln_approx_aux2_def: fixes x' :: "real"   and e :: "real"   and n :: "nat"   and y :: "real" shows "ln_approx_aux2 x' e n y = ln_approx_aux3 x' e n y (y ^ (2 * n + 1) / (1 - y\<^sup>2) / real (2 * n + 1))"
bernoulli_stream_pref_prob': fixes M :: "bool stream measure"   and p :: "real"   and n :: "nat"   and x :: "bool stream" assumes "M = bernoulli_stream p"   and "p \<le> 1"   and "0 \<le> p" shows "emeasure M {w \<in> space M. stake n w = stake n x} = ennreal (prod (prob_component p x) {0..<n})"
free: shows "(x, \<tau>) \<notin> FV \<Gamma>"
components_fst_sym_preproc_addnew_subset: fixes gs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and vs :: "'t list"   and fs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and v :: "'t" shows "component_of_term ` set (fst (sym_preproc_addnew gs vs fs v)) \<subseteq> component_of_term ` (Keys (set gs) \<union> insert v (set vs))"
measure_nonneg': fixes M :: "'c measure"   and A :: "'c set" shows "\<not> Sigma_Algebra.measure M A < 0"
pr0_ext: fixes a :: "'a"   and b :: "'a" assumes "\<not> (ide a \<and> ide b)" shows "\<pp>\<^sub>0[a, b] = null"
iter_winden_step_ivl_termination: fixes c :: "com" shows "\<exists>C. iter_widen (step_ivl \<top>) (bot c) = Some C"
indRelRTPO_impl_TRel_is_weak_reduction_simulation: fixes TRel :: "('procT \<times> 'procT) set" assumes "weak_reduction_simulation (indRelRTPO TRel) (STCal Source Target)" shows "weak_reduction_simulation (TRel\<^sup>+) Target"
perp_down_in_S: fixes a :: "proj2"   and b :: "proj2"   and l :: "proj2_line" assumes "a \<in> hyp2"   and "b \<in> hyp2"   and "proj2_incident b l" shows "perp_down a l \<in> S"
foundation4: fixes \<tau> :: "'b state \<times> 'b state" shows "\<tau> |\<noteq> null"
visits_via_ltl: fixes A :: "'a set"   and W :: "'a set" assumes "visits_via P' A W"   and "v0 \<notin> W" shows "visits_via (ltl P') A W"
bs3: fixes t1 :: "lam"   and t2 :: "lam"   and t3 :: "lam" assumes " t1 \<longrightarrow>\<^sub>\<beta>\<^sup>* t2"   and " t2 \<longrightarrow>\<^sub>\<beta>\<^sup>* t3" shows " t1 \<longrightarrow>\<^sub>\<beta>\<^sup>* t3"
P_Some: fixes z :: "'a option" assumes "\<And>x. z = Some x \<Longrightarrow> (P \<circ> Some) x" shows "P z"
eval_monom_smallomega': fixes b :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list"   and e :: "real"   and es :: "real list" assumes "basis_wf (b # basis)"   and "e < 0" shows "eval_monom (1, es) basis \<in> \<omega>(\<lambda>x. b x powr e)"
bind_map_option: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c option"   and g :: "'b \<Rightarrow> 'a option" shows "Option.bind (map_option f x) g = Option.bind x (g \<circ> f)"
cod_cod: fixes f :: "'a" assumes "L.arr f" shows "L.cod (L.cod f) = L.cod f"
gauge_Int: fixes \<gamma>1 :: "'a \<Rightarrow> 'a set"   and \<gamma>2 :: "'a \<Rightarrow> 'a set" assumes "gauge \<gamma>1"   and "gauge \<gamma>2" shows "gauge (\<lambda>x. \<gamma>1 x \<inter> \<gamma>2 x)"
assms_2: fixes x :: "'b"   and y :: "'b" shows "f (x + y) = f x + f y"
add_right_cancel: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b"   and c :: "'b" assumes "class.cancel_semigroup_add plus" shows "(plus b a = plus c a) = (b = c)"
1: fixes y :: "'a stream"   and z :: "'b stream" shows "PROP P (y ||| z)"
span_clauses_4: fixes x :: "'a"   and S :: "'a set"   and c :: "'a" assumes "x \<in> More_Modules.ideal S" shows "c * x \<in> More_Modules.ideal S"
assms_3: fixes j :: "nat" assumes "hi < j"   and "j < length xs" shows "xs' ! j = xs ! j"
extremeD_2: fixes X :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and e :: "'b"   and x :: "'b" assumes "extreme X r e"   and "x \<in> X" shows "r x e"
assms: shows "a_consistent (ae, a, as) (\<Gamma>, Bool b, Alts e1 e2 # S)"
bij_betw_prod_divisors_coprime: fixes a :: "nat"   and b :: "nat" assumes "coprime a b" shows "bij_betw (\<lambda>x. fst x * snd x) ({d. d dvd a} \<times> {d. d dvd b}) {k. k dvd a * b}"
impl_alt_def: fixes P :: "'a upred"   and Q :: "'a upred" shows "(P \<Rightarrow> Q) = (\<not> P \<or> Q)"
seqr_post_out: fixes r :: "('b, 'c) urel"   and P :: "'b hrel"   and Q :: "('b, 'c) urel" assumes "in\<alpha> \<sharp> r" shows "P ;; (Q \<and> r) = (P ;; Q \<and> r)"
supportE_preTz: fixes C :: "acom"   and e :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> nat"   and n :: "nat" shows "supportE (\<lambda>l. preTz C (e l) n) \<subseteq> supportE e"
entries_RBT_Impl_diag: fixes t :: "('c, 'd) rbt" shows "RBT_Impl.entries (RBT_Impl_diag t) = map (\<lambda>(k, v). ((k, k), v)) (RBT_Impl.entries t)"
mono_sync2b: assumes "P \<sqsubseteq> Q" shows "\<forall>r. r \<in> min_elems (D (P \<lbrakk>A\<rbrakk> S)) \<longrightarrow> r \<in> T (Q \<lbrakk>A\<rbrakk> S)"
Diff_Diff_Int: fixes A :: "'e set"   and B :: "'e set" shows "A - (A - B) = A \<inter> B"
check_eq_dist_nodes: fixes M :: "nat bdd list \<times> bool list"   and T :: "bool list list"   and m :: "nat"   and v :: "nat"   and l :: "nat bdd"   and r :: "nat bdd" assumes "\<forall>p q. dfa_is_node M q \<and> p < q \<longrightarrow> tr_lookup T q p = (\<exists>n<m. dist_nodes M n v p q)"   and "0 < m"   and "bdd_all (dfa_is_node M) l"   and "bdd_all (dfa_is_node M) r" shows "(\<not> check_eq l r T) = (\<exists>bs. bddh (length bs) l \<and> bddh (length bs) r \<and> (\<exists>n<m. dist_nodes M n v (bdd_lookup l bs) (bdd_lookup r bs)))"
coprime_divisors: fixes a :: "'b"   and c :: "'b"   and b :: "'b"   and d :: "'b" assumes "a dvd c"   and "b dvd d"   and "algebraic_semidom_class.coprime c d" shows "algebraic_semidom_class.coprime a b"
Bind_def: fixes g :: "'b \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and ab :: "bool" shows "Bind g f ab \<equiv> \<lambda>P s. let a = g s in f a ab P s"
if_elim: fixes b :: "'BExp"   and p :: "('Var, 'AExp, 'BExp) Stmt"   and q :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>Stmt.If b p q, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>" obtains "c' = p" and "mem' = mem" and "mds' = mds" and "ev\<^sub>B mem b"   | "c' = q" and "mem' = mem" and "mds' = mds" and "\<not> ev\<^sub>B mem b"
assms: shows "update rt dip (dsn, dsk, flg, hops, sip, {}) \<noteq> rt"
finite_rderiv0: fixes a :: "'a" shows "finite {fold rderiv xs (FBase a) |xs. True}"
closed_except_def: fixes size :: "'a \<Rightarrow> nat"   and frees :: "'a \<Rightarrow> name fset"   and subst :: "'a \<Rightarrow> (name, 'a) fmap \<Rightarrow> 'a"   and "consts" :: "'a \<Rightarrow> name fset"   and app :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unapp :: "'a \<Rightarrow> ('a \<times> 'a) option"   and const :: "name \<Rightarrow> 'a"   and unconst :: "'a \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'a"   and unfree :: "'a \<Rightarrow> name option"   and t :: "'a"   and S :: "name fset" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.closed_except frees t S = (frees t |\<subseteq>| S)"
not_arr_null: shows "\<not> hhom\<^sub>D.arr hhom\<^sub>D.null"
simps_6: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> ('c, 'b) Term.term list \<Rightarrow> 'a"   and x21 :: "'c"   and x22 :: "('c, 'b) Term.term list" shows "(case Fun x21 x22 of Var x \<Rightarrow> f1 x | Fun x xa \<Rightarrow> f2 x xa) = f2 x21 x22"