hcju/isabelleps huggingface dataset:
actions_ok_iff: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b"   and ta :: "('a, 'b, 'e, 'c, 'd, 'f) thread_action" shows "actions_ok s t ta = (lock_ok_las (locks s) t \<lbrace>ta\<rbrace>\<^bsub>l\<^esub> \<and> thread_oks (thr s) \<lbrace>ta\<rbrace>\<^bsub>t\<^esub> \<and> cond_action_oks s t \<lbrace>ta\<rbrace>\<^bsub>c\<^esub> \<and> wset_actions_ok (wset s) t \<lbrace>ta\<rbrace>\<^bsub>w\<^esub> \<and> interrupt_actions_ok (interrupts s) \<lbrace>ta\<rbrace>\<^bsub>i\<^esub>)"
getParts_foldr: fixes rs :: "'a simple_rule list" shows "getParts rs = foldr partIps (extract_IPSets rs) [wordinterval_UNIV]"
reflection: shows "rel_reflects_binary_pred (indRelRTPO TRel) (\<lambda>b. hasBarb b (STCalWB SWB TWB))"
order_gt_0_iff_finite: shows "(0 < order G) = finite (carrier G)"
distrib_4: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and y :: "'b"   and z :: "'b"   and x :: "'b" assumes "class.distrib_lattice inf less_eq less sup" shows "inf (sup y z) x = sup (inf y x) (inf z x)"
simps_1: fixes ts :: "t list"   and ts' :: "t list" shows "c_types_agree (Type ts) ts' = (ts = ts')"
dims_A'_1: fixes ws :: "nat \<Rightarrow> real" shows "dim_row (A' ws) = prod_list (nths (dims (A ws)) (Collect even))"
borel_measurable_subalgebra: fixes N :: "'b measure"   and M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" assumes "sets N \<subseteq> sets M"   and "space N = space M"   and "f \<in> borel_measurable N" shows "f \<in> borel_measurable M"
assms_5: shows "Y \<in> P1 \<union> P2"
infinite_greaterThan: fixes n :: "nat" shows "infinite {n<..}"
length_filter_conv_card: fixes p :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "length (filter p xs) = card {i. i < length xs \<and> p (xs ! i)}"
Isubst_supp: fixes t1 :: "trmI"   and x :: "name"   and t2 :: "trmI" shows "supp (t1[x::=t2]) \<subseteq> supp t2 \<union> (supp t1 - {x})"
first_ind_length: fixes x :: "'b"   and xs :: "'b list"   and i :: "nat" assumes "x \<in> set xs" shows "first_ind xs x i < length xs + i"
good_item_simps_35: fixes xsa :: "'varSort"   and xa :: "'var"   and delta :: "'e"   and inp :: "'c \<Rightarrow> ('c, 'd, 'varSort, 'var, 'e) qTerm set option"   and binpa :: "'d \<Rightarrow> ('c, 'd, 'varSort, 'var, 'e) qAbs set option" shows "Var xsa xa \<noteq> Op delta inp binpa"
strong_low_bisim_mm_def: fixes \<R> :: "((('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) \<times> ('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) set" shows "strong_low_bisim_mm \<R> \<equiv> sym \<R> \<and> closed_glob_consistent \<R> \<and> (\<forall>c\<^sub>1 mds mem\<^sub>1 c\<^sub>2 mem\<^sub>2. (\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle>, \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>) \<in> \<R> \<longrightarrow> mem\<^sub>1 =\<^bsub>mds\<^esub>\<^sup>l mem\<^sub>2 \<and> (\<forall>c\<^sub>1' mds' mem\<^sub>1'. \<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<leadsto> \<langle>c\<^sub>1', mds', mem\<^sub>1'\<rangle> \<longrightarrow> (\<exists>c\<^sub>2' mem\<^sub>2'. \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle> \<leadsto> \<langle>c\<^sub>2', mds', mem\<^sub>2'\<rangle> \<and> (\<langle>c\<^sub>1', mds', mem\<^sub>1'\<rangle>, \<langle>c\<^sub>2', mds', mem\<^sub>2'\<rangle>) \<in> \<R>)))"
inf_ccSUP: fixes B :: "'c set"   and a :: "'a"   and f :: "'c \<Rightarrow> 'a" assumes "countable B" shows "inf a (Sup (f ` B)) = (SUP b\<in>B. inf a (f b))"
eq: shows "map_poly to_fract p = smult c' (map_poly to_fract e)"
equality: fixes r :: "'a control_state_scheme"   and r' :: "'a control_state_scheme" assumes "break_status r = break_status r'"   and "return_status r = return_status r'"   and "more r = more r'" shows "r = r'"
fls_subdegree_minus: fixes f :: "'a fls"   and g :: "'a fls" assumes "f - g \<noteq> 0" shows "min (fls_subdegree f) (fls_subdegree g) \<le> fls_subdegree (f - g)"
simps_6: fixes x :: "nat"   and P :: "pol"   and y :: "nat"   and Q :: "pol" shows "Pinj x P \<langle>*\<rangle> Pinj y Q = (if x = y then mkPinj x (P \<langle>*\<rangle> Q) else if y < x then mkPinj y (Pinj (x - y) P \<langle>*\<rangle> Q) else mkPinj x (Pinj (y - x) Q \<langle>*\<rangle> P))"
Int_Un_eq_1: fixes S :: "'c set"   and T :: "'c set" shows "S \<inter> T \<union> S = S"
of_rat_2_2: fixes x :: "rat" shows "invariant_2 (of_rat_2 x)"
less_mset_zmsetE: fixes M :: "'b zmultiset"   and N :: "'b zmultiset" assumes "M < 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 < B"
rep_eq: fixes x :: "'b fset set" shows "fset (Inf x) = (if finite (\<Inter> (fset ` x)) then \<Inter> (fset ` x) else {})"
ereal_uminus_greaterThan: fixes a :: "ereal" shows "uminus ` {a<..} = {..<- a}"
VEnd_polymap: fixes p :: "'f poly" shows "VEnd (polymap p)"
icd_is_the_icd: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and i :: "nat"   and \<pi> :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "IFC prob"   and "IFC_def.is_icdi prob i \<pi> k" shows "k = (THE k. IFC_def.is_icdi prob i \<pi> k)"
Wbis_MtransC: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com"   and s :: "'state"   and t :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state" assumes "c \<approx>w d"   and "s \<approx> t"   and "(c, s) \<rightarrow>*c (c', s')" obtains d' :: "('test, 'atom) com" and t' :: "'state" where "(d, t) \<rightarrow>*c (d', t'__)" and "s' \<approx> t'__" and "c' \<approx>w d'"   | t' :: "'state" where "(d, t) \<rightarrow>*t t'" and "s' \<approx> t'" and "discr c'"
hchop_def: fixes v :: "view"   and u :: "view"   and w :: "view" shows "v=u\<parallel>w \<equiv> R_Chop(ext v,ext u,ext w) \<and> lan v = lan u \<and> lan v = lan w \<and> own v = own u \<and> own v = own w \<and> more v = more w \<and> more v = more u"
iso_dghm_ObjMap_vrange: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>i\<^sub>s\<^sub>o\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<R>\<^sub>\<circ> (\<FF>\<lparr>ObjMap\<rparr>) = \<BB>\<lparr>Obj\<rparr>"
lc_map_scale: fixes c :: "'b"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "lcf (c \<cdot> p) = c * lcf p"
ca_ex: shows "c \<otimes> a = somegcd G (c \<otimes> a) (c \<otimes> b) \<otimes> x__"
nonzero_root_set_finite: fixes les :: "(real \<times> real \<times> real) set" assumes "finite les" shows "finite (nonzero_root_set les)"
that: fixes a :: "'a" assumes "a \<in> span S"   and "a \<noteq> (0::'a)"   and "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> a \<bullet> x" shows "thesis"
2: shows "x * - z\<^sup>\<star> \<sqinter> z\<^sup>+ = bot"
a1: shows "s' = snd (fst (sub_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0"
simps: fixes index :: "nat"   and b :: "rat vec" shows "vars_from_index_geq_vec index b = map (\<lambda>i. GEQ (lp_monom 1 (i + index)) (b $ i)) [0..<dim_vec b]"
right: shows "revision_step r' s\<^sub>1 s\<^sub>2'"
mapCollectE: fixes x :: "'a"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and m :: "'b \<Rightarrow> 'c option" assumes "x \<in> mapCollect f m" obtains k :: "'b" and v :: "'c" where "m k = Some v" and "x = f k v"
id_prs: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and Abs :: "'e \<Rightarrow> 'f"   and Rep :: "'f \<Rightarrow> 'e" assumes "Quotient3 R Abs Rep" shows "(Rep ---> Abs) id = id"
subst_3: fixes x :: "name"   and y :: "name"   and t' :: "lam"   and t :: "lam" assumes "x \<sharp> (y, t')" shows "Lam [x].t[y::=t'] = Lam [x].t[y::=t']"
simps_1: fixes nrm :: "lname set"   and brk :: "label \<Rightarrow> lname set"   and more :: "'a" shows "nrm \<lparr>nrm = nrm, brk = brk, \<dots> = more\<rparr> = nrm"
iter0: fixes c :: "'a::refinement_lattice" shows "(c::'a::refinement_lattice)\<^sup>\<omega> \<sqsubseteq> nil"
1: shows "Broadcast (i, Insert e n) \<in> set (history ja)"
delta_sol: shows "F.delta = lr (dual F.delta, F.delta)"
\<eta>o_def: fixes y :: "'b" shows "Adj'.\<eta>o y = \<phi>' y (F y)"
K_cfg_cfg_onI: fixes cfg :: "'s cfg"   and x :: "'s"   and cfg' :: "'s cfg" assumes "cfg \<in> cfg_on x"   and "cfg' \<in> set_pmf (K_cfg cfg)" shows "cfg' \<in> cfg_on (state cfg')"
map_ident: fixes t :: "'a msllist" shows "mslmap (\<lambda>x. x) t = t"
assign_named: fixes A :: "'b set"   and H :: "(('c, 'b) fm list \<times> 'b) set"   and i :: "'b"   and a :: "'b" assumes "Hintikka A H"   and "i \<in> proj (hequiv_rel A H) a" shows "i \<in> names H"
height_balL2: fixes l :: "('b \<times> nat) tree"   and r :: "('b \<times> nat) tree"   and a :: "'b" assumes "avl l"   and "avl r"   and "height l \<noteq> height r + 2" shows "height (balL l a r) = 1 + max (height l) (height r)"
simps_12: fixes \<nu> :: "'b"   and va :: "'b E.term"   and vab :: "'b E.term" shows "\<^bold>\<langle>\<nu>\<^bold>\<rangle> \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> va \<^bold>\<star> vab = \<^bold>\<langle>\<nu>\<^bold>\<rangle> \<^bold>\<star> va \<^bold>\<star> vab"
mon_pl_ileq: fixes w :: "('a set \<times> 'a set) list"   and w' :: "('a set \<times> 'a set) list" assumes "w \<preceq> w'" shows "mon_pl w \<subseteq> mon_pl w'"
assms: shows "(u, v) \<in> stutter_extend_edges V E"
infdist_le: fixes a :: "'c"   and A :: "'c set"   and x :: "'c" assumes "a \<in> A" shows "infdist x A \<le> dist x a"
add_mset_diff_bothsides: fixes a :: "'b"   and M :: "'b multiset"   and A :: "'b multiset" shows "(a, M) - (a, A) = M - A"
IH: fixes ves :: "e list" assumes "ves = map (Basic \<circ> EConst) vs" shows "const_list ves"
ereal_inverse_nonneg_iff: fixes x :: "ereal" shows "(0 \<le> inverse x) = (0 \<le> x \<or> x = - \<infinity>)"
max_def: fixes a :: "'b"   and b :: "'b" shows "ord_class.max a b = (if a \<le> b then b else a)"
old_next_limit_finite: fixes \<phi> :: "'e ltlr" shows "finite (old_next_limit \<phi>)"
eq_onp_same_args: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b" shows "eq_onp P x x = P x"
valid_lists_cc_restr: fixes S :: "Vars.var set"   and G :: "CoCalls" shows "valid_lists S G = valid_lists S (G G|` S)"
ProcRec1: fixes \<Gamma> :: "'b \<Rightarrow> ('c, 'b, 'd) com option"   and \<Theta> :: "('c set \<times> 'b \<times> 'c set \<times> 'c set) set"   and P :: "'a \<Rightarrow> 'c set"   and p :: "'b"   and Q :: "'a \<Rightarrow> 'c set"   and A :: "'a \<Rightarrow> 'c set"   and F :: "'d set" assumes "\<forall>Z. \<Gamma>,\<Theta> \<union> (\<Union>Z. {(P Z, p, Q Z, A Z)})\<turnstile>\<^bsub>/F \<^esub>(P Z) the (\<Gamma> p) (Q Z),(A Z)"   and "p \<in> dom \<Gamma>" shows "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P Z) Call p (Q Z),(A Z)"
lm032: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and B :: "'d set" assumes "finite A" shows "sum f A = sum f (A \<inter> B) + sum f (A - B)"
ll: shows "measure_pmf.prob (bernoulli_pmf p \<bind> (\<lambda>b. if b then map_pmf Inl Da else map_pmf Inr Db)) {f. \<exists>e. Inr e = f} = 1 - p"
distinct_17: fixes act :: "act"   and ccs :: "ccs"   and ccs' :: "ccs" shows "act.ccs \<noteq> !ccs'"
preserves_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<Rightarrow> b\<guillemotright>" shows "\<guillemotleft>S.UP f : S.UP a \<Rightarrow>\<^sub>S S.UP b\<guillemotright>"
Quotient_rep_equiv2: fixes R :: "'j \<Rightarrow> 'j \<Rightarrow> bool"   and Abs :: "'j \<Rightarrow> 'k"   and Rep :: "'k \<Rightarrow> 'j"   and T :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and a :: "'j"   and b :: "'k" assumes "Quotient R Abs Rep T"   and "T a b" shows "R (Rep b) a"
assms: shows "\<Gamma> \<turnstile>g gpv \<surd>"
adjoint_adjoint: fixes A :: "'a mat" shows "adjoint (adjoint A) = A"
rel: shows "P \<sim>\<lbrakk>\<cdot>\<rbrakk><SRel,TRel> Q"
assms_2: fixes i :: "nat" assumes "i < length xs" shows "R (xs ! i) (f (xs ! i) (ys ! i))"
standard_decomp_append: fixes k :: "nat"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" assumes "standard_decomp k ps"   and "standard_decomp k qs" shows "standard_decomp k (ps @ qs)"
qSwap_compose: fixes zs' :: "'f"   and x' :: "'g"   and y' :: "'g"   and zs :: "'f"   and x :: "'g"   and y :: "'g"   and X :: "('d, 'e, 'f, 'g, 'h) qTerm" shows "X #[[x \<and> y]]_zs #[[x' \<and> y']]_zs' = X #[[x' \<and> y']]_zs' #[[x @zs[x' \<and> y']_zs' \<and> y @zs[x' \<and> y']_zs']]_zs"
execute_parallel_operator_positive_effect_if: fixes s :: "'a \<Rightarrow> bool option"   and ops :: "'a strips_operator list"   and op :: "'a strips_operator"   and v :: "'a" assumes "are_all_operators_applicable s ops"   and "are_all_operator_effects_consistent ops"   and "op \<in> set ops"   and "v \<in> set (add_effects_of op)" shows "execute_parallel_operator s ops v = Some True"
assms_1: shows "filterlim f G (at x within {x. P x})"
lset_ltl: fixes xs :: "'c llist" shows "lset (ltl xs) \<subseteq> lset xs"
le_less_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<preceq> y"   and "y \<prec> z" shows "x \<prec> z"
permute_eq_iff: fixes p :: "perm"   and x :: "'b"   and y :: "'b" shows "(p \<bullet> x = p \<bullet> y) = (x = y)"
suf_sing_power: fixes w :: "'b list"   and a :: "'b"   and m :: "nat" assumes "w \<le>s [a] \<^sup>@ m" shows "w \<in> [a]*"
Rep_eqD: fixes x :: "'b word"   and y :: "'b word" assumes "uint x = uint y" shows "x = y"
m1: shows "tensor_P (1\<^sub>m N) M1 * tensor_P (1\<^sub>m N) M1 = tensor_P (1\<^sub>m N) M1"
real_sqrt_eq_zero_cancel: fixes x :: "real" assumes "0 \<le> x"   and "sqrt x = 0" shows "x = 0"
nat_pow_Suc2: fixes x :: "'a mod_ring poly"   and n :: "nat" assumes "x \<in> carrier R" shows "x [^]\<^bsub>R\<^esub> Suc n = x \<otimes>\<^bsub>R\<^esub> x [^]\<^bsub>R\<^esub> n"
fun_eq_on_def: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and A :: "'b set" shows "fun_eq_on f g A \<equiv> \<forall>a\<in>A. f a = g a"
satisfies_atom'_def: fixes v :: "nat \<Rightarrow> 'a"   and a :: "'a atom" shows "v \<Turnstile>\<^sub>a\<^sub>e a = (v (atom_var a) = atom_const a)"
field_derivative_eq_vector_derivative: fixes f :: "real \<Rightarrow> real"   and x :: "real" shows "deriv f x = vector_derivative f (at x)"
simps: fixes G :: "('a, 'b) graph_scheme"   and a :: "'a list" shows "cycle G a = (\<exists>v p. a = p \<and> path G v p v \<and> p \<noteq> [] \<and> distinct p)"
power_int_mono_iff: fixes a :: "'a"   and b :: "'a"   and n :: "int" assumes "(0::'a) \<le> a"   and "(0::'a) \<le> b"   and "0 < n" shows "(a powi n \<le> b powi n) = (a \<le> b)"
necess_cond_1_lhs: fixes x :: "'a" assumes "x \<in> \<V>" shows "int (size {#p \<in># mset_set (\<V> - {x}) \<times># filter_mset ((\<in>) x) \<B>. fst p \<in> snd p#}) = \<B> rep x * (\<k> - 1)"
normal_invE_1: fixes N :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set set" assumes "N \<lhd> relative_homology_group (int n) (nsphere n) (equator n)" shows "subgroup N (relative_homology_group (int n) (nsphere n) (equator n))"
fst_pair: fixes a :: "'a"   and b :: "'b" shows "fst (a, b) = a"
ALGxy_det_def: fixes A :: "(nat \<times> nat list) list"   and qs :: "'c list"   and init :: "'c list"   and x :: "'c"   and y :: "'c" shows "ALGxy_det A qs init x y = (\<Sum>i<length qs. if qs ! i \<in> {y, x} then ALG'_det A qs init i y + ALG'_det A qs init i x else 0)"
straighten_downtop_def: fixes x :: "wall"   and y :: "wall" shows "straighten_downtop x y \<equiv> x = basic [cup, vert] \<circ> basic [vert, cap] \<and> y = basic [vert] \<circ> basic [vert]"
conf_clinit_diff: fixes P :: "jvm_prog"   and sh :: "sheap"   and stk :: "val list"   and loc :: "val list"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and stk' :: "val list"   and loc' :: "val list"   and pc' :: "nat" assumes "conf_clinit P sh ((stk, loc, C, M, pc, ics) # frs)" shows "conf_clinit P sh ((stk', loc', C, M, pc', ics) # frs)"
Exit_source: fixes a :: "'b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node" assumes "lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit a"   and "src a = NewExit" shows "False"
rep_\<R>_ex: fixes R :: "('c \<Rightarrow> real) set" assumes "R \<in> \<R>" shows "(SOME u. u \<in> R) \<in> R"
wf\<^sub>s\<^sub>s\<^sub>t_append_suffix': fixes V :: "'e set"   and S :: "('f, 'e) stateful_strand_step list"   and S' :: "('f, 'e) stateful_strand_step list" assumes "wf'\<^sub>s\<^sub>s\<^sub>t V S"   and "(\<Union>x\<in>set S'. case x of receive\<langle>x\<rangle> \<Rightarrow> fv x | \<langle>assign: x \<doteq> t'\<rangle> \<Rightarrow> fv t' | \<langle>check: x \<doteq> t'\<rangle> \<Rightarrow> {} | insert\<langle>t,t'\<rangle> \<Rightarrow> fv t \<union> fv t' | _ \<Rightarrow> {}) \<subseteq> wfvarsoccs\<^sub>s\<^sub>s\<^sub>t S \<union> V" shows "wf'\<^sub>s\<^sub>s\<^sub>t V (S @ S')"
Field: shows "Field (Restr r B) = Field r \<inter> B"
proj2_line_abs_def: fixes v :: "(real, 3) vec" shows "proj2_line_abs v \<equiv> P2L (proj2_abs v)"
assms_5: assumes "shd (stl s) = shd s" shows "P"
fun_upd_same_oexp: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and x :: "'b"   and y :: "'a" assumes "(f, g) \<in> oexp"   and "f x = g x"   and "x \<in> Field s"   and "y \<in> Field r" shows "(f(x := y), g(x := y)) \<in> oexp"
swap_mu_cost_def: fixes dmu :: "int iarray iarray"   and i :: "nat"   and dmu_i_im1 :: "int"   and dim1 :: "int"   and di :: "int"   and dsi :: "int" shows "swap_mu_cost dmu i dmu_i_im1 dim1 di dsi = (let im1 = i - 1; res = IArray.of_fun (\<lambda>ii. if ii < im1 then dmu !! ii else if i < ii then let dmu_ii = dmu !! ii in IArray.of_fun (\<lambda>j. let dmu_ii_j = dmu_ii !! j in if j = i then (dsi * dmu_ii !! im1 - dmu_i_im1 * dmu_ii_j) div di else if j = im1 then (dmu_i_im1 * dmu_ii_j + dmu_ii !! i * dim1) div di else dmu_ii_j) ii else if ii = i then let mu_im1 = dmu !! im1 in IArray.of_fun (\<lambda>j. if j = im1 then dmu_i_im1 else mu_im1 !! j) ii else IArray.of_fun ((!!) (dmu !! i)) ii) m in Let (8 * (m - Suc i)) (Pair res))"
GB_replace_red_rtranclp_stable_GB_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "dickson_grading d"   and "G \<subseteq> dgrad_p_set d m"   and "is_Groebner_basis G"   and "p \<in> G"   and "(red (G - {p}))\<^sup>*\<^sup>* p q" shows "is_Groebner_basis (insert q (G - {p}))"
polyfun_rootbound: fixes c :: "nat \<Rightarrow> 'a"   and k :: "nat"   and n :: "nat" assumes "c k \<noteq> (0::'a)"   and "k \<le> n" shows "finite {z. (\<Sum>i\<le>n. c i * z ^ i) = (0::'a)} \<and> card {z. (\<Sum>i\<le>n. c i * z ^ i) = (0::'a)} \<le> n"
assoc_in_hom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b arrow_of_spans_data"   and g :: "'b arrow_of_spans_data"   and h :: "'b arrow_of_spans_data" assumes "span_bicategory C prj0 prj1"   and "partial_magma.ide (span_vertical_category.vcomp C) f"   and "partial_magma.ide (span_vertical_category.vcomp C) g"   and "partial_magma.ide (span_vertical_category.vcomp C) h"   and "span_vertical_category.src C f = span_vertical_category.trg C g"   and "span_vertical_category.src C g = span_vertical_category.trg C h" shows "partial_magma.in_hom (span_vertical_category.vcomp C) \<lparr>Chn = three_composable_identity_arrows_of_spans.chine_assoc C prj0 prj1 f g h, Dom = Dom (span_bicategory.hcomp C prj0 prj1 (span_bicategory.hcomp C prj0 prj1 f g) h), Cod = Cod (span_bicategory.hcomp C prj0 prj1 f (span_bicategory.hcomp C prj0 prj1 g h))\<rparr> (span_bicategory.hcomp C prj0 prj1 (span_bicategory.hcomp C prj0 prj1 f g) h) (span_bicategory.hcomp C prj0 prj1 f (span_bicategory.hcomp C prj0 prj1 g h))"
selectlike_co_selectlike: fixes X :: "(variable \<Rightarrow> real) set"   and \<nu> :: "variable \<Rightarrow> real"   and V :: "variable set" shows "selectlike (- selectlike X \<nu> V) \<nu> V = selectlike (- X) \<nu> V"
node_disconnectTE: fixes ii :: "nat"   and s :: "'c"   and R :: "nat set"   and i :: "nat"   and i' :: "nat"   and s' :: "'c"   and R' :: "nat set"   and S :: "('c \<times> 'd seq_action \<times> 'c) set" assumes "(NodeS ii s R, disconnect(i, i'), NodeS ii s' R') \<in> node_sos S" obtains "i = ii" and "s' = s" and "R' = R - {i'}"   | "i' = ii" and "s' = s" and "R' = R - {i}"   | "ii \<noteq> i" and "ii \<noteq> i'" and "s' = s" and "R' = R"
simps_2: fixes a :: "nat"   and b :: "nat" shows "swap_fm a b FalseF = FalseF"
at_set_avoiding_aux: fixes Xs :: "atom set"   and As :: "atom set" assumes "Xs \<subseteq> As"   and "finite As" shows "\<exists>p. p \<bullet> Xs \<inter> As = {} \<and> supp p = Xs \<union> p \<bullet> Xs"
a: shows "M[a\<turnstile>c>b] \<longrightarrow>\<^sub>c M'"
check_simplex': fixes s :: "QDelta ns_constraint list \<times> (('a, ('a \<times> QDelta atom) list) mapping \<times> ((nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping) \<times> 'a list) \<times> ('a, QDelta) state" shows "check_simplex (Simplex_State s) = map_sum id Simplex_State (check_simplex' s)"
less_eq_pred_def: fixes P :: "'a pred"   and Q :: "'a pred" shows "(P \<le> Q) = (eval P \<le> eval Q)"
left_neutral: fixes a :: "'d multiset" shows "{#} \<union># a = a"
double_dual_iso_def: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "finite_comm_group G" shows "finite_comm_group.double_dual_iso G x = (\<lambda>\<chi>. if character G \<chi> then \<chi> x else 0)"
exprel_refl: fixes X :: "freeExp" shows "X \<sim> X"
tame13a_def: fixes g :: "graph" shows "tame13a g = (\<exists>w. admissible w g \<and> ListSum (faces g) w < squanderTarget)"
subcategory_axioms: shows "subcategory (\<cdot>) (left g)"
indRelL_def: shows "indRelL \<equiv> _Collect (x, y) (indRelLp x y)"
echelon_form_upt_row_condition1_explicit: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat"   and i :: "'c" assumes "echelon_form_upt_row A k"   and "mod_type_class.to_nat i < k"   and "is_zero_row i A" shows "\<not> (\<exists>j>i. mod_type_class.to_nat j < k \<and> \<not> is_zero_row j A)"
simps_2: fixes x :: "'a"   and xs :: "'a list"   and f :: "'a \<Rightarrow> nat"   and l :: "nat"   and up :: "nat"   and left :: "nat" shows "lsteps (x # xs) f l up left = (if f x < l then lsteps xs f l up left else if l < f x then up + lsteps (x # xs) f (l + 1) up left else left + lsteps xs f l up left)"
fls_left_inverse_eq_fls_right_inverse: fixes x :: "'a"   and f :: "'a fls"   and y :: "'a" assumes "x * fls_nth f (fls_subdegree f) = (1::'a)"   and "fls_nth f (fls_subdegree f) * y = (1::'a)" shows "fls_left_inverse f x = fls_right_inverse f y"
ennreal_top_neq_of_nat: fixes i :: "nat" shows "\<top> \<noteq> of_nat i"
orbit_stab_fun_is_inj: fixes x :: "'c"   and R1 :: "'a set"   and R2 :: "'a set" assumes "x \<in> E"   and "R1 \<in> rcosets stabilizer G \<phi> x"   and "R2 \<in> rcosets stabilizer G \<phi> x"   and "\<phi> (inv (SOME h. h \<in> R1)) x = \<phi> (inv (SOME h. h \<in> R2)) x" shows "R1 = R2"
arcosh_cosh_real: fixes x :: "real" assumes "0 \<le> x" shows "arcosh (cosh x) = x"
lemma_poly_linear_rem: fixes h :: "'a"   and t :: "'a list"   and a :: "'a" shows "\<exists>q r. h # t = [r] +++ [- a, 1::'a] *** q"
semiring_norm_90: fixes m :: "num"   and n :: "num" shows "(num.Bit1 m = num.Bit1 n) = (m = n)"
blinding_of_on_discrete: shows "blinding_of_on UNIV id (=)"
connected_iff3: fixes x :: "'a" assumes "is_inj x"   and "is_p_fun x" shows "connected x = (x\<^sup>T ; 1 ; x \<le> x\<^sup>\<star> + x\<^sup>T\<^sup>\<star>)"
prefix_snoc: fixes xs :: "'d list"   and ys :: "'d list"   and y :: "'d" shows "prefix xs (ys @ [y]) = (xs = ys @ [y] \<or> prefix xs ys)"