hcju/isabelleps huggingface dataset:
valid_path_split_2: 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"   and as' :: "'a list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.valid_path kind get_return_edges (as @ as')" shows "CFG.valid_path kind get_return_edges as'"
P: shows "P \<subseteq> A\<^sup>\<spadesuit>"
fw_out_of_bounds2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "nat"   and j' :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and k :: "nat"   and i' :: "nat" assumes "class.linordered_ab_monoid_add plus zero less_eq less"   and "n < j'" shows "linordered_ab_monoid_add.fw plus less_eq M n k i' j' = M i' j'"
finite2: shows "(\<Sum>\<^sup>+ x. \<integral>\<^sup>+y\<in>X. (h_diff i (x, y) * indicator X x)\<partial>count_space UNIV) \<noteq> \<top>"
Var_BetasD: fixes x :: "name"   and t :: "Lambda_Z.term" assumes "Var x \<rightarrow>\<^sub>\<beta>\<^sup>* t" shows "t = Var x"
I_to_outer: fixes S :: "'a list"   and B :: "nat list"   and I :: "'a \<Rightarrow> node_state option"   and P :: "(nat \<times> 'a set) list"   and D :: "'a set" assumes "((S, B, I, P), [], D, {}) \<in> GS_rel" shows "(I, D) \<in> oGS_rel"
restrict0_in_fun_space: fixes f :: "'c \<Rightarrow> real" assumes "diff_fun k charts_eucl f" shows "restrict0 manifold_eucl.dest.carrier f \<in> manifold_eucl.dest.diff_fun_space"
simps_1: shows "used [] = (\<Union>B. parts (initState B))"
comp_in_homI: fixes f :: "('a S.E.term, 'a) S.arr"   and a :: "('a S.E.term, 'a) S.arr"   and b :: "('a S.E.term, 'a) S.arr"   and g :: "('a S.E.term, 'a) S.arr"   and c :: "('a S.E.term, 'a) S.arr" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>S b\<guillemotright>"   and "\<guillemotleft>g : b \<Rightarrow>\<^sub>S c\<guillemotright>" shows "\<guillemotleft>g \<cdot>\<^sub>S f : a \<Rightarrow>\<^sub>S c\<guillemotright>"
remdups_adj_append': fixes xs :: "'g list"   and ys :: "'g list" assumes "xs = [] \<or> ys = [] \<or> last xs \<noteq> hd ys" shows "remdups_adj (xs @ ys) = remdups_adj xs @ remdups_adj ys"
sel_2: fixes x1 :: "heap"   and x2 :: "nat set" shows "addrOf (pHeap x1 x2) = x2"
flow_continuous_on_compose: fixes C :: "'b set"   and g :: "'b \<Rightarrow> 'a"   and h :: "'b \<Rightarrow> real" assumes "continuous_on C g"   and "continuous_on C h"   and "\<And>c. c \<in> C \<Longrightarrow> g c \<in> X"   and "\<And>c. c \<in> C \<Longrightarrow> h c \<in> existence_ivl0 (g c)" shows "continuous_on C (\<lambda>c. flow0 (g c) (h c))"
REM_gives_desired_upper_bound: shows "backflows (set (filter_compliant_stateful_ACS G M edgesList)) - edges G - (backflows (set (filter_compliant_stateful_ACS G M edgesList)) - backflows (set (filter_IFS_no_violations G M edgesList)) - edges G) = backflows (set (filter_compliant_stateful_ACS G M edgesList) \<inter> set (filter_IFS_no_violations G M edgesList)) - edges G"
A: shows "A \<subseteq> \<Union> (B ` S)"
inDed_def: fixes xs :: "formula list" shows "inDed xs = (xs \<in> deductions CutFreePC)"
simps_2: fixes a :: "'b"   and as :: "'b list" shows "Poly (a # as) = pCons a (Poly as)"
not_coprime_cong_eq_0: fixes p :: "int"   and a :: "int" assumes "prime p"   and "\<not> coprime a p" shows "[a = 0] (mod p)"
fv\<^sub>s\<^sub>s\<^sub>t_def: fixes S :: "('b, 'a) stateful_strand_step list" shows "fv\<^sub>s\<^sub>s\<^sub>t S \<equiv> \<Union> (set (map fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p S))"
orthogonal_complement_of_closure: fixes A :: "'b set" shows "orthogonal_complement A = orthogonal_complement (closure A)"
E_3: fixes n :: "nat" shows "outer_measure_of M (A n) = emeasure M (E n)"
sync_def: fixes P :: "'b process"   and A :: "'b set"   and Q :: "'b process" shows "P \<lbrakk>A\<rbrakk> Q \<equiv> Abs_process ({(s, R). \<exists>t u X Y. (t, X) \<in> F P \<and> (u, Y) \<in> F Q \<and> s setinterleaves ((t, u), ev ` A \<union> {tick}) \<and> R = (X \<union> Y) \<inter> (ev ` A \<union> {tick}) \<union> X \<inter> Y} \<union> {(s, R). \<exists>t u r v. front_tickFree v \<and> (tickFree r \<or> v = []) \<and> s = r @ v \<and> r setinterleaves ((t, u), ev ` A \<union> {tick}) \<and> (t \<in> D P \<and> u \<in> T Q \<or> t \<in> D Q \<and> u \<in> T P)}, {s. \<exists>t u r v. front_tickFree v \<and> (tickFree r \<or> v = []) \<and> s = r @ v \<and> r setinterleaves ((t, u), ev ` A \<union> {tick}) \<and> (t \<in> D P \<and> u \<in> T Q \<or> t \<in> D Q \<and> u \<in> T P)})"
adjointFO_ode_vars: fixes I :: "('sf, 'sc, 'sz) interp"   and \<sigma> :: "'b \<Rightarrow> ('sf, 'sz) trm"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and ODE :: "('sf + 'b, 'sz) ODE"   and \<omega> :: "(real, 'sz) vec \<times> (real, 'sz) vec" shows "ODE_vars (adjointFO I \<sigma> \<nu>) ODE = ODE_vars (adjointFO I \<sigma> \<omega>) ODE"
hseqE: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "hseq \<nu> \<mu>" obtains "arr \<mu>" and "arr \<nu>" and "src \<nu> = trg \<mu>"
vconn_implies_hpar_1: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and \<mu> :: "'c"   and f :: "'c"   and g :: "'c" assumes "horizontal_homs V src trg"   and "partial_magma.in_hom V \<mu> f g" shows "src \<mu> = src f"
of_real_eq_Quat: fixes a :: "real" shows "quat_of_real a = Quat a 0 0 0"
Complete_subset_Gen: fixes I :: "item set"   and k :: "nat"   and u :: "nat" assumes "I \<subseteq> Gen (\<P> k u)"   and "k \<le> length Doc" shows "Complete k I \<subseteq> Gen (\<P> k u)"
Mp_square_free_m: fixes m :: "int"   and f :: "int poly" shows "poly_mod.square_free_m m (poly_mod.Mp m f) = poly_mod.square_free_m m f"
assms_6: shows "\<forall>x. x \<in> ChSetP \<longrightarrow> x \<in> loc PQ"
unit_simps_2: fixes a :: "'a" assumes "C.obj a" shows "src\<^sub>D (F.unit a) = F.map\<^sub>0 a"
mult_mono_nonneg_nonpos: fixes c :: "'a"   and a :: "'a"   and b :: "'a"   and d :: "'a" assumes "c \<le> a"   and "b \<le> d"   and "(0::'a) \<le> a"   and "d \<le> (0::'a)" shows "a * b \<le> c * d"
commit_for_sc_in_\<E>: fixes E :: "('thread_id \<times> ('addr, 'thread_id) obs_event action) llist"   and ws :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "E \<in> \<E>"   and "P \<turnstile> (E, ws) \<surd>"   and "sequentially_consistent P (E, ws)" shows "justifying_exec (commit_for_sc P (E, ws) n) \<in> \<E>"
distance_upper_bound: fixes v :: "'a"   and xs :: "'a list"   and w :: "'a" assumes "v \<leadsto>xs\<leadsto>\<^bsub>H\<^esub> w" shows "H.distance v w \<le> length xs"
polytope_imp_closed: fixes S :: "'a set" assumes "polytope S" shows "closed S"
dghm_dghm_comp_dghm_id_right: shows "\<FF> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M dghm_id \<AA> = \<FF>"
run_snth: fixes A :: "('c, 'd) ngba"   and r :: "('c \<times> 'd) stream"   and p :: "'d"   and k :: "nat" assumes "run A r p" shows "fst (r !! k) \<in> alphabet A \<and> snd (r !! k) \<in> transition A (fst (r !! k)) (target (stake k r) p)"
strong_tricolour_inv_def: shows "strong_tricolour_inv = (\<lambda>s. \<forall>x xa. black x s \<longrightarrow> \<not> ((x points_to xa) s \<and> white xa s))"
that: fixes i :: "'a" assumes "i \<in> Basis" shows "continuous_on X (\<lambda>x. blinfun_apply (f x) i)"
iso_is_image_block: fixes \<V> :: "'c set"   and \<B> :: "'c set multiset"   and \<V>' :: "'d set"   and \<B>' :: "'d set multiset"   and bij_map :: "'c \<Rightarrow> 'd" assumes "incidence_system_isomorphism \<V> \<B> \<V>' \<B>' bij_map" shows "\<B>' = blocks_image \<B> bij_map"
ordst_trans: fixes X :: "char list"   and Y :: "char list"   and Z :: "char list" assumes "ordst X Y"   and "ordst Y Z" shows "ordst X Z"
l3_inv4I: fixes x :: "l3_state" assumes "Tags \<subseteq> ik x" shows "x \<in> l3_inv4"
unat_minus_one: fixes w :: "'b word" assumes "w \<noteq> 0" shows "unat (w - 1) = unat w - 1"
dominant_term_ms_aux_MSLCons: fixes C :: "'a"   and e :: "real"   and xs :: "('a \<times> real) msllist" shows "dominant_term_ms_aux (MSLCons (C, e) xs) = map_prod id ((#) e) (dominant_term C)"
p2r_id_neg: fixes p :: "'a \<Rightarrow> bool" shows "Id \<inter> - \<lceil>p\<rceil> = \<lceil>- p\<rceil>"
maxsimp_vertices: fixes w :: "'w"   and s :: "'w" assumes "w \<in> W"   and "s \<in> S" shows "w +o \<langle>S - {s}\<rangle> \<in> dual_order.poset_simplex_map \<P> {w}"
star_isotone: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "x\<^sup>\<star> \<le> y\<^sup>\<star>"
simps_2: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c1 :: "('b, 'c) com"   and c2 :: "('b, 'c) com" assumes "PL_Indis indis" shows "PL_Indis.SC_siso tval aval indis (c1 ;; c2) = (PL_Indis.SC_siso tval aval indis c1 \<and> PL_Indis.SC_siso tval aval indis c2)"
subset_hull: fixes S :: "'c set \<Rightarrow> bool"   and t :: "'c set"   and s :: "'c set" assumes "S t" shows "(S hull s \<subseteq> t) = (s \<subseteq> t)"
comp_fun_commute_funpow: fixes g :: "'a \<Rightarrow> nat" shows "comp_fun_commute_on (\<lambda>x. f x ^^ g x) A"
af_letter_congruent: fixes eq :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> bool"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and \<nu> :: "'b set" assumes "af_congruent eq"   and "eq \<phi> \<psi>" shows "eq (af_letter \<phi> \<nu>) (af_letter \<psi> \<nu>)"
tab_simps_2: shows "src \<mu> = src u\<^sub>0"
ci_stk_offerI: fixes ci :: "'a instr \<Rightarrow> (nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog \<Rightarrow> 'b \<Rightarrow> 'a val list \<Rightarrow> 'a val list \<Rightarrow> String.literal \<Rightarrow> String.literal \<Rightarrow> nat \<Rightarrow> 'a frame list \<Rightarrow> bool" assumes "\<And>ins P h stk stk' loc C M pc frs. ci ins P h stk loc C M pc frs \<Longrightarrow> ci ins P h (stk @ stk') loc C M pc frs" shows "ci_stk_offer ci"
upset_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "ord.upset less_eq = ord.upset_set less_eq \<circ> \<eta>"
presSwapAbs_def: fixes hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'c"   and MOD :: "('index, 'bindex, 'varSort, 'd, 'opSym, 'var, 'e, 'c, 'f) Recursion.model_scheme" shows "presSwapAbs hA MOD \<equiv> \<forall>zs z1 z2 us s A. wlsAbs (us, s) A \<longrightarrow> hA (A $[z1 \<and> z2]_zs) = gSwapAbs MOD zs z1 z2 A (hA A)"
distinguished_bounded_support: fixes x :: "('idx, 'pred, 'act) formula"   and P :: "'state"   and Q :: "'state" assumes "x distinguishes P from Q" obtains y :: "('idx, 'pred, 'act) formula" where "supp y \<subseteq> supp P" and "y distinguishes P from Q"
lmeasurable_ball: fixes a :: "'c"   and r :: "real" shows "ball a r \<in> lmeasurable"
L_infinity_AE_ebound: fixes f :: "'b \<Rightarrow> real"   and M :: "'b measure" shows "AE x in M. ennreal \<bar>f x\<bar> \<le> eNorm (\<LL> \<infinity> M) f"
expl_cond_exp_borel: fixes Y :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure"   and X :: "'b \<Rightarrow> real" assumes "Y \<in> space M \<rightarrow> space N"   and "disc_fct Y"   and "\<forall>r\<in>range Y \<inter> space N. \<exists>A\<in>sets N. range Y \<inter> A = {r}" shows "expl_cond_expect M Y X \<in> borel_measurable (fct_gen_subalgebra M N Y)"
len: shows "length xs' = length ys + 1 + length ws + 1 + length zs"
embed_bool_idem: fixes P :: "'a \<Rightarrow> bool"   and s :: "'a" shows "\<guillemotleft> P \<guillemotright> s * \<guillemotleft> P \<guillemotright> s = \<guillemotleft> P \<guillemotright> s"
assms: shows "low_equal s1 s2 \<and> t1 = snd (fst (set_annul True (snd (fst (write_cpu (cpu_reg_val PC s2 + sign_ext24 (ucast (get_operand_w22 (snd instr ! Suc 0)) << 2)) nPC (snd (fst (write_cpu (cpu_reg_val nPC s2) PC s1)))))))) \<and> t2 = snd (fst (set_annul True (snd (fst (write_cpu (cpu_reg_val PC s2 + sign_ext24 (ucast (get_operand_w22 (snd instr ! Suc 0)) << 2)) nPC (snd (fst (write_cpu (cpu_reg_val nPC s2) PC s2))))))))"
fpxs_radical_def: fixes rt :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and r :: "nat"   and f :: "'b fpxs" shows "fpxs_radical rt r f = (if f = 0 then 0 else let f' = fls_base_factor_to_fps (fls_of_fpxs f); f'' = fpxs_of_fls (fps_to_fls (fps_radical rt r f')) in fpxs_shift (- fpxs_val f / rat_of_nat r) (fpxs_compose_power f'' (1 / rat_of_nat (fpxs_root_order f))))"
map2_val_compat'D3: fixes f :: "('a \<times> 'd) list \<times> 'o \<Rightarrow> ('a \<times> 'e) list \<times> 'o"   and zs :: "('a \<times> 'd) list \<times> 'o" assumes "map2_val_compat' f"   and "oalist_inv zs" shows "fst ` set (fst (f zs)) \<subseteq> fst ` set (fst zs)"
decomposable_function_subst: fixes f :: "'a ltl \<Rightarrow> 'a ltl"   and \<phi> :: "'a ltl" assumes "f true = true"   and "f false = false"   and "\<And>\<phi> \<psi>. f (\<phi> and \<psi>) = f \<phi> and f \<psi>"   and "\<And>\<phi> \<psi>. f (\<phi> or \<psi>) = f \<phi> or f \<psi>" shows "f \<phi> = subst \<phi> (\<lambda>\<chi>. Some (f \<chi>))"
compl_le_swap2: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and y :: "'a"   and x :: "'a" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top"   and "less_eq (uminus y) x" shows "less_eq (uminus x) y"
l3_dy_payload_refines_dy_fake_msg: fixes M :: "msg" assumes "M \<in> payload" shows "{R23s \<inter> UNIV \<times> l3_inv5} l2_dy_fake_msg M, l3_dy M {> R23s}"
lhd_wit: fixes xs :: "'a llist" assumes "chain R\<^sup>+\<^sup>+ xs" shows "lhd (local.wit xs) = lhd xs"
run_altc_sample_option: fixes altc_sample :: "'b \<Rightarrow> ('d \<Rightarrow> 'm) \<Rightarrow> 'm"   and p :: "'b"   and f :: "'d \<Rightarrow> ('a, 'm) optionT" shows "run_option (local.altc_sample_option altc_sample p f) = altc_sample p (\<lambda>x. run_option (f x))"
\<Sigma>_positive: fixes j :: "int"   and l :: "int"   and f :: "'d \<Rightarrow> 'c" assumes "j < l" shows "\<Sigma> f j l = sum (f \<circ> of_int) {j..<l}"
filter_chan_set_hd_tr_filter: fixes l :: "'c list"   and cs :: "'c set" shows "tr_filter l cs \<noteq> [] \<longrightarrow> filter_chan_set (hd (tr_filter l cs)) cs"
column_of_matrix_def: fixes A :: "'b matrix"   and n :: "nat" shows "column_of_matrix A n \<equiv> take_columns (move_matrix A 0 (- int n)) 1"
diffeomorphism_add: fixes k :: "enat"   and c :: "'c" shows "Smooth.diffeomorphism k UNIV UNIV (\<lambda>x. x + c) (\<lambda>x. x - c)"
mycasexyx_def: fixes x :: "'a"   and y :: "'a" shows "mycasexyx x y = Plus (seq [Star (Plus (Atom x) (Atom y)), Atom x, Atom y, Atom x]) (seq [Star (Plus (Atom x) (Atom y)), Atom y, Atom x, Atom y])"
neg: shows "pairwise (\<lambda>S S'. negligible (S \<inter> S')) \<T>"
path_app: fixes \<alpha>e :: "'a \<Rightarrow> ('b \<times> 'c \<times> 'b) set"   and \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and g :: "'a"   and ns :: "'b list"   and ms :: "'b list" assumes "graph_path \<alpha>e \<alpha>n invar inEdges'"   and "graph_path_base.path \<alpha>n invar inEdges' g ns"   and "graph_path_base.path \<alpha>n invar inEdges' g ms"   and "last ns = hd ms" shows "graph_path_base.path \<alpha>n invar inEdges' g (ns @ tl ms)"
is_CSP_processE1: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" assumes "is_CSP_process P" obtains "P is CSP1 healthy" and "P is CSP2 healthy" and "P is R healthy"
Class_closed: fixes a :: "'a"   and b :: "'a" assumes "a \<in> Class b"   and "b \<in> G" shows "a \<in> G"
can_occur_def: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and ev :: "('d, 'e, 'f) event"   and c :: "('d, 'e, 'f) configuration" assumes "distributed_system channel" shows "distributed_system.can_occur channel trans send recv ev c \<equiv> case ev of Trans p s s' \<Rightarrow> states c p = s \<and> trans p s s' | Send i p q s s' msg \<Rightarrow> states c p = s \<and> channel i = Some (p, q) \<and> send i p q s s' msg | Recv i p q s s' msg \<Rightarrow> states c p = s \<and> channel i = Some (q, p) \<and> 0 < length (msgs c i) \<and> hd (msgs c i) = Msg msg \<and> recv i p q s s' msg | Snapshot p \<Rightarrow> \<not> ps c p \<noteq> None | RecvMarker i p q \<Rightarrow> channel i = Some (q, p) \<and> 0 < length (msgs c i) \<and> hd (msgs c i) = Marker"
B: shows "xps @ yps = c_tr step out s\<^sub>0 (map fst (xps @ yps))"
filterlim_iff_le_filtercomap: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'c filter"   and G :: "'b filter" shows "filterlim f F G = (G \<le> filtercomap f F)"
rcosetsI: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and x :: "'c" assumes "Group.group G"   and "H \<subseteq> carrier G"   and "x \<in> carrier G" shows "H #>\<^bsub>G\<^esub> x \<in> rcosets\<^bsub>G\<^esub> H"
IH_2: fixes \<sigma> :: "('a \<times> 'a trm) list"   and \<eta> :: "('a \<times> 'a trm) list" assumes "equivalent_on \<sigma> \<eta> (vars_of t2) I" shows "I (t2 \<lhd> \<sigma>) (t2 \<lhd> \<eta>)"
subst_comp_all: fixes \<delta> :: "'c \<Rightarrow> ('a, 'd) Term.term"   and \<theta> :: "'d \<Rightarrow> ('a, 'b) Term.term"   and M :: "('a, 'c) Term.term set" shows "M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<delta> \<circ>\<^sub>s \<theta> = M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<delta> \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>"
Vs_3: shows "R.Span K (Vs @ Us) = E"
eucl_le_atLeast: fixes a :: "'c" shows "{x. \<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i} = {a..}"
that: fixes CAs :: "'a literal multiset list"   and AAs :: "'a multiset list"   and As :: "'a list"   and E :: "'a literal multiset" assumes "set CAs \<subseteq> N"   and "INTERP N \<Turnstile>m mset CAs"   and "\<And>CA. CA \<in> set CAs \<Longrightarrow> production N CA \<noteq> {}"   and "ord_resolve CAs DA AAs As E"   and "\<not> INTERP N \<Turnstile> E"   and "E < DA" shows "thesis"
invertible_imp_matrix_inv: fixes A :: "(('b, 'c) vec, 'c) vec" assumes "invertible A" shows "matrix_inv A = (1::'b) div det A *k adjugate A"
eq_iff: fixes a :: "'t \<Rightarrow>\<^sub>0 'c"   and b :: "'t \<Rightarrow>\<^sub>0 'c" shows "(a = b) = (b \<preceq>\<^sub>p a \<and> a \<preceq>\<^sub>p b)"
simplex_index_2: fixes cs :: "('b \<times> constraint) list"   and v :: "(nat, rat) mapping" assumes "simplex_index cs = Inr v" shows "\<langle>v\<rangle> \<Turnstile>\<^sub>c\<^sub>s (snd ` set cs)"
float_le_less_trans: fixes a :: "('a, 'b) IEEE.float"   and b :: "('a, 'b) IEEE.float"   and c :: "('a, 'b) IEEE.float" assumes "is_finite a"   and "is_finite b"   and "is_finite c"   and "a \<le> b"   and "b < c" shows "a < c"
Lemma1_1: fixes x :: "ValD" shows "\<eta>: x \<mapsto> Lub DtoKM_i\<cdot>x"
poincare_line_sym_general: fixes u :: "complex_homo"   and v :: "complex_homo" assumes "u \<noteq> v"   and "u \<noteq> inversion v" shows "poincare_line u v = poincare_line v u"
1: shows "p0 \<cdot> induced_arrow (C.dom p0') (D.mkCone p0' p1') = p0'"
connectedin_empty: fixes X :: "'b topology" shows "connectedin X {}"
iso_lunit: fixes f :: "'b" assumes "D.ide f" shows "D.iso \<l>\<^sub>D[f]"
last_index_Cons: fixes x :: "'b"   and xs :: "'b list"   and y :: "'b" shows "last_index (x # xs) y = (if x = y then if x \<in> set xs then last_index xs y + 1 else 0 else last_index xs y + 1)"
comp_func: fixes f :: "int \<Rightarrow> int" assumes "f \<in> CofGroups.Ex1" shows "upOne \<circ> f \<in> CofGroups.Ex1"
etc_1: shows "h\<^sub>0 \<bottom> h\<^sub>1"
method_typing_aux_3: shows "method_typing_aux x xa xb xc xd"
runit_eqI: fixes f :: "'a"   and \<mu> :: "'a" assumes "ide f"   and "\<guillemotleft>\<mu> : f \<star> src f \<Rightarrow> f\<guillemotright>"   and "\<mu> \<star> src f = (f \<star> \<i>[src f]) \<cdot> \<a>[f, src f, src f]" shows "\<mu> = \<r>[f]"
w0_def: shows "P'.w0 \<equiv> lhd (ltl P')"
of_int_floor_cancel: fixes x :: "'a" shows "(of_int \<lfloor>x\<rfloor> = x) = (\<exists>n. x = of_int n)"
assms_2: shows "set \<phi>s \<subseteq> fmla"
arity_var_def: fixes n :: "nat" shows "arity_var n = \<infinity>"
better_NotL_substn: fixes y :: "name"   and M :: "trm"   and a :: "coname"   and c :: "coname"   and P :: "trm" assumes "y \<sharp> M" shows "NotL <a>.M y{y:=<c>.P} = fresh_fun (\<lambda>x'. Cut <c>.P x'.NotL <a>.M x')"
h_3: fixes p :: "nat"   and X :: "'a topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int" assumes "singular_chain p X c" shows "singular_chain (Suc p) X (h p c)"
rep_eq: fixes x :: "'d"   and xa :: "'e"   and xb :: "('d, 'e) mapping" shows "mapping.rep (Mapping.update x xa xb) = mapping.rep xb(x \<mapsto> xa)"
at_right_minus: fixes a :: "real" shows "at_right a = filtermap uminus (at_left (- a))"
crel_vs1_alt_def: fixes R :: "'f \<Rightarrow> 'g \<Rightarrow> bool" shows "crel_vs1 R = state_dp_consistency.crel_vs R OO local.rel_state (=)"
tagged_partial_division_ofD_3: fixes s :: "('a \<times> 'a set) set"   and i :: "'a set"   and x :: "'a"   and K :: "'a set" assumes "s tagged_partial_division_of i"   and "(x, K) \<in> s" shows "K \<subseteq> i"
simps_5: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "abort \<Rightarrow> 'a"   and x1 :: "'b" shows "(case Rraise x1 of Rraise x \<Rightarrow> f1 x | Rabort x \<Rightarrow> f2 x) = f1 x1"
bij_lens_axioms_def: fixes x :: "'a \<Longrightarrow> 'b" shows "bij_lens_axioms x \<equiv> \<forall>\<sigma> \<rho>. put\<^bsub>x\<^esub> \<sigma> (get\<^bsub>x\<^esub> \<rho>) = \<rho>"
lemPlaneContainsBasePoint: fixes pl :: "'a Plane" shows "inPlane (pbasepoint pl) pl"
Im_sum: fixes f :: "'a \<Rightarrow> complex"   and s :: "'a set" shows "Im (sum f s) = (\<Sum>x\<in>s. Im (f x))"
all_def: fixes l :: "'b list"   and f :: "'b \<Rightarrow> bool" shows "all l f = list_all f l"
hd_def: fixes list :: "'b list" shows "hd list = (case list of x21 # x22 \<Rightarrow> x21)"
exhaust: fixes y :: "nonce_t" obtains x1 :: "fresh_t" where "y = nonce_fresh x1"   | x2 :: "nat" where "y = nonce_atk x2"
distinct_11: fixes x3 :: "Vars.var"   and x4 :: "Vars.var" shows "Upd x3 \<noteq> Dummy x4"
summable_strong: fixes z :: "complex" assumes "cmod z < 1" shows "summable (\<lambda>n. (a gchoose n) * z ^ n)"
match_def: fixes \<Delta> :: "'state \<Rightarrow> 'b \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool"   and s :: "'state"   and s1 :: "'state"   and vl1 :: "'value list"   and a :: "'act"   and ou :: "'out"   and s' :: "'state"   and vl' :: "'b" shows "match \<Delta> s s1 vl1 a ou s' vl' \<equiv> \<exists>a1 ou1 s1' vl1'. let trn = Trans s a ou s'; trn1 = Trans s1 a1 ou1 s1' in validTrans trn1 \<and> consume trn1 vl1 vl1' \<and> \<gamma> trn = \<gamma> trn1 \<and> (\<gamma> trn \<longrightarrow> g trn = g trn1) \<and> \<Delta> s' vl' s1' vl1'"
lQuot_map_project: fixes as :: "'a"   and n :: "nat"   and A :: "'a list set" assumes "as \<in> \<Sigma> n"   and "A \<subseteq> lists (\<Sigma> (Suc n))" shows "lQuot as (map project ` A) = map project ` (\<Union>a\<in>set (embed as). lQuot a A)"
pCons_eq_iff: fixes a :: "'d"   and p :: "'d poly"   and b :: "'d"   and q :: "'d poly" shows "(pCons a p = pCons b q) = (a = b \<and> p = q)"
h3g2a: shows "\<forall>f. sum f {..1} = f 0 + f 1"
majority_nonempty: fixes S :: "Disk set" assumes "IsMajority S" shows "S \<noteq> {}"
arev_dom: fixes a :: "'b" shows "(a \<in> arcs G) = (rev_G a \<noteq> a)"
dom_eqI: fixes a :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "H.VxV.ide a"   and "H.VxV.seq f a" shows "H.VxV.dom f = a"