hcju/isabelleps huggingface dataset:
CK_nf_set: shows "{f. CK_nf f} = {id, C, K, C \<circ> K, K \<circ> C, C \<circ> K \<circ> C, K \<circ> C \<circ> K, C \<circ> K \<circ> C \<circ> K, K \<circ> C \<circ> K \<circ> C, C \<circ> K \<circ> C \<circ> K \<circ> C, K \<circ> C \<circ> K \<circ> C \<circ> K, C \<circ> K \<circ> C \<circ> K \<circ> C \<circ> K, K \<circ> C \<circ> K \<circ> C \<circ> K \<circ> C, C \<circ> K \<circ> C \<circ> K \<circ> C \<circ> K \<circ> C}"
l4_16R3: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p"   and D' :: "'p" assumes "A B C D FSC A' B' C' D'"   and "A \<noteq> B"   and "Bet C A B" shows "Cong C D C' D'"
real_cond_exp_F_meas: fixes f :: "'a \<Rightarrow> real" assumes "integrable M f"   and "f \<in> borel_measurable Invariants" shows "AE x in M. real_cond_exp M Invariants f x = f x"
power_odd_eq: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and n :: "nat" assumes "class.monoid_mult one times" shows "power.power one times a (Suc (2 * n)) = times a (power.power one times (power.power one times a n) 2)"
rep_dlist_def: shows "rep_dlist \<equiv> quot_type.rep Rep_dlist"
comp_inv_arr: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "VoV.inverse_arrows f g" shows "VoV.comp g f = VoV.dom f"
ntr_split: fixes ca :: "'n list multiset"   and cb :: "'n list multiset"   and w :: "('p, 'ba) label list list"   and c' :: "'n list multiset" assumes "(ca + cb, w, c') \<in> trcl (ntr fg)"   and "valid fg (ca + cb)" shows "\<exists>ca' cb' waa wb. c' = ca' + cb' \<and> w \<in> waa \<otimes>\<^bsub>\<alpha>n fg\<^esub> wb \<and> mon_c fg ca \<inter> (mon_c fg cb \<union> mon_ww fg wb) = {} \<and> mon_c fg cb \<inter> (mon_c fg ca \<union> mon_ww fg waa) = {} \<and> (ca, waa, ca') \<in> trcl (ntr fg) \<and> (cb, wb, cb') \<in> trcl (ntr fg)"
sorted_sorted_wrt: shows "signed.sorted = sorted_wrt (\<le>s)"
card_order_cprod: fixes r1 :: "('k \<times> 'k) set"   and r2 :: "('l \<times> 'l) set" assumes "card_order r1"   and "card_order r2" shows "card_order (r1 *c r2)"
algebraic_def: fixes x :: "'b" shows "algebraic x = (\<exists>p. (\<forall>i. coeff p i \<in> \<int>) \<and> p \<noteq> 0 \<and> poly p x = (0::'b))"
lfp_induct: fixes f :: "'a \<Rightarrow> 'a"   and P :: "'a" assumes "mono f"   and "f (inf (lfp f) P) \<le> P" shows "lfp f \<le> P"
inext_nth_append_eq1: fixes A :: "nat set"   and B :: "nat set"   and n :: "nat" assumes "finite A"   and "A \<noteq> {}"   and "Max A < iMin B"   and "A \<rightarrow> n \<noteq> Max A" shows "(A \<union> B) \<rightarrow> n = A \<rightarrow> n"
CFG_SSA_def: fixes \<alpha>e :: "'d \<Rightarrow> ('e \<times> 'f \<times> 'e) set"   and \<alpha>n :: "'d \<Rightarrow> 'e list"   and invar :: "'d \<Rightarrow> bool"   and inEdges' :: "'d \<Rightarrow> 'e \<Rightarrow> ('e \<times> 'f) list"   and Entry :: "'d \<Rightarrow> 'e"   and defs :: "'d \<Rightarrow> 'e \<Rightarrow> 'g set"   and "uses" :: "'d \<Rightarrow> 'e \<Rightarrow> 'g set"   and phis :: "'d \<Rightarrow> 'e \<times> 'g \<Rightarrow> 'g list option" shows "CFG_SSA \<alpha>e \<alpha>n invar inEdges' Entry defs uses phis \<equiv> CFG \<alpha>e \<alpha>n invar inEdges' Entry defs uses \<and> CFG_SSA_axioms \<alpha>n inEdges' defs phis"
the_coext_unique: fixes x' :: "'d"   and f :: "'c"   and g :: "'d" assumes "arrow_from_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F x' x f"   and "x\<epsilon>.is_coext x' f g" shows "g = x\<epsilon>.the_coext x' f"
mulex_on_union_right: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and F :: "'b set"   and A :: "'b multiset"   and B :: "'b multiset"   and K :: "'b multiset" assumes "mulex_on P F A B"   and "K \<in> multisets F" shows "mulex_on P F A (K + B)"
simps: fixes vo :: "'d option"   and kts :: "('c \<times> ('c, 'd) trie) list" shows "invar_trie (Trie vo kts) = (distinct (map fst kts) \<and> (\<forall>(k, t)\<in>set kts. \<not> is_empty_trie t \<and> invar_trie t))"
refine: fixes G :: "('a, 'b) b_graph_rec_scheme" shows "blue_dfs_fe G \<equiv> FOREACH\<^sub>C (g_V0 G) (\<lambda>(uu_, uu_, cyc). cyc = NO_CYC) (\<lambda>v0 (blues, reds, uu_). if v0 \<notin> blues then REC\<^sub>T (blue_dfs_body G) (blues, reds, {}, v0) \<bind> (\<lambda>(blues, reds, uu_, cyc). RETURN (blues, reds, cyc)) else RETURN (blues, reds, NO_CYC)) ({}, {}, NO_CYC)"
irreducible_normalized_divisors: fixes x :: "'a"   and y :: "'a" assumes "irreducible x"   and "y dvd x"   and "normalize y = y" shows "y = (1::'a) \<or> y = normalize x"
graph_extD1: fixes H :: "'b set"   and h :: "'b \<Rightarrow> real"   and H' :: "'b set"   and h' :: "'b \<Rightarrow> real"   and x :: "'b" assumes "graph H h \<subseteq> graph H' h'"   and "x \<in> H" shows "h x = h' x"
hyps: shows "r \<down>m r|\<sigma>| = r \<down>l \<sigma>"
permutes_inj: fixes p :: "'b \<Rightarrow> 'b"   and S :: "'b set" assumes "p permutes S" shows "inj p"
nth_default_Bernstein_coeffs: fixes P :: "real poly"   and p :: "nat"   and c :: "real"   and d :: "real"   and i :: "nat" assumes "degree P \<le> p" shows "nth_default 0 (Bernstein_coeffs p c d P) i = inverse (real (p choose i)) * coeff (reciprocal_poly p (P \<circ>\<^sub>p [:c, 1:] \<circ>\<^sub>p [:0, d - c:]) \<circ>\<^sub>p [:1, 1:]) (p - i)"
hd_in_set: fixes xs :: "'d list" assumes "xs \<noteq> []" shows "hd xs \<in> set xs"
substn_subst_0: fixes t :: "dB"   and s :: "dB" shows "substn t s 0 = t[s/0]"
mult_le_cancel_right: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and abs :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and c :: "'e"   and b :: "'e" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less_eq (times a c) (times b c) = ((less zero c \<longrightarrow> less_eq a b) \<and> (less c zero \<longrightarrow> less_eq b a))"
WhileE: fixes b :: "bexp"   and c :: "com"   and s :: "char list \<Rightarrow> int"   and t :: "char list \<Rightarrow> int" assumes "(WHILE b DO c, s) \<Rightarrow> t" obtains "t = s" and "\<not> bval b s"   | s\<^sub>2 :: "char list \<Rightarrow> int" where "bval b s" and "(c, s) \<Rightarrow> s\<^sub>2" and "(WHILE b DO c, s\<^sub>2) \<Rightarrow> t"
3: shows "\<guillemotleft>e'e.trnr\<^sub>\<eta> m \<theta> : m \<Rightarrow> src w \<star> e'\<guillemotright>"
degree3_coeffs: fixes p :: "'b poly" assumes "degree p = 3" shows "\<exists>a b c d. p = [:d, c, b, a:] \<and> a \<noteq> (0::'b)"
range_subsetD: fixes f :: "'d \<Rightarrow> 'c"   and B :: "'c set"   and i :: "'d" assumes "range f \<subseteq> B" shows "f i \<in> B"
elims_2: fixes x :: "symbol list"   and xa :: "(nat \<times> symbol \<times> symbol list) list"   and xb :: "symbol list" assumes "LeftDerivation x xa xb" obtains a :: "symbol list" and b :: "symbol list" where "x = aa__" and "xa = []" and "xb = ba__" and "aa__ = ba__"   | a :: "symbol list" and d :: "nat \<times> symbol \<times> symbol list" and D :: "(nat \<times> symbol \<times> symbol list) list" and b :: "symbol list" where "x = a" and "xa = d # D" and "xb = b" and "\<exists>x. LeftDerives1 a (fst d) (snd d) x \<and> LeftDerivation x D b"
plus_MkI_MkI: fixes x :: "int"   and y :: "int" shows "MkI\<cdot>x + MkI\<cdot>y = MkI\<cdot>(x + y)"
order_iff_strict: fixes a :: "'b"   and b :: "'b" shows "(a \<le> b) = (a < b \<or> a = b)"
HEndPhase0_HInv4d_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase0 s s' p"   and "HInv4d s p"   and "Inv2c s"   and "Inv1 s" shows "HInv4d s' p"
M_SMP_repr: shows "finite_SMP_representation arity Ana \<Gamma> M"
simps_87: fixes f1 :: "'d \<Rightarrow> 'a"   and f2 :: "'e \<Rightarrow> 'b"   and f3 :: "'f \<Rightarrow> 'c"   and x7 :: "('d, 'e, 'f) expr" shows "map_expr f1 f2 f3 (Rfork x7) = Rfork (map_expr f1 f2 f3 x7)"
neg_le_divideR_eq: fixes c :: "real"   and a :: "'a"   and b :: "'a" assumes "c < 0" shows "(a \<le> b /\<^sub>R c) = (b \<le> c *\<^sub>R a)"
vderiv_on_continuous_on: fixes f :: "real \<Rightarrow> 'b"   and f' :: "real \<Rightarrow> 'b"   and S :: "real set" assumes " D f = f' on S" shows "continuous_on S f"
assms_1: shows "A \<approx>\<^sub>\<circ> B"
aB_def: fixes v :: "Value" shows "aB v = (case v of boolV b \<Rightarrow> b)"
formulaTrueAppendValuation: fixes formula :: "Literal list list"   and valuation :: "Literal list"   and valuation' :: "Literal list" assumes "formulaTrue formula valuation" shows "formulaTrue formula (valuation @ valuation')"
elims_3: fixes outPorts :: "'a \<Rightarrow> 'b fset"   and x :: "'a" assumes "\<not> Vertex_Graph.terminal_node outPorts x" obtains n :: "'a" where "x = n" and "outPorts n \<noteq> {||}"
lprefix_antisym: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "lprefix xs ys"   and "lprefix ys xs" shows "xs = ys"
catch_gpv_Done: fixes x :: "'d" shows "catch_gpv (Generative_Probabilistic_Value.Done x) = Generative_Probabilistic_Value.Done (Some x)"
mult_right: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> o[F](g)" shows "(\<lambda>x. f x * h x) \<in> o[F](\<lambda>x. g x * h x)"
split_indicator: fixes P :: "'b \<Rightarrow> bool"   and S :: "'d set"   and x :: "'d" shows "P (indicator S x) = ((x \<in> S \<longrightarrow> P (1::'b)) \<and> (x \<notin> S \<longrightarrow> P (0::'b)))"
\<pi>_bigtheta: shows "\<pi> \<in> \<Theta>(\<lambda>x. x / ln x)"
wf_iff_no_infinite_down_chain: fixes r :: "('a \<times> 'a) set" shows "wf r = (\<nexists>f. \<forall>i. (f (Suc i), f i) \<in> r)"
Var_eq_apps_conv: fixes m :: "nat"   and s :: "dB"   and ss :: "dB list" shows "(Var m = s \<degree>\<degree> ss) = (Var m = s \<and> ss = [])"
add_assoc: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "a \<sqinter> b \<sqinter> c = a \<sqinter> (b \<sqinter> c)"
poly_MVT': fixes a :: "real"   and b :: "real"   and A :: "real set"   and p :: "real poly" assumes "{min a b..max a b} \<subseteq> A" shows "\<exists>x\<in>A. poly p b - poly p a = (b - a) * poly (pderiv p) x"
linear_scale: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec"   and r :: "'a"   and b :: "('a, 'b) vec" assumes "Vector_Spaces.linear (*s) (*s) f" shows "f (r *s b) = r *s f b"
oconf_fupd: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list"   and F :: "char list"   and T :: "ty"   and D :: "char list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and v :: "val"   and fs :: "char list \<times> char list \<Rightarrow> val option" assumes "P \<turnstile> C has F:T in D"   and "P,h \<turnstile> v :\<le> T"   and "P,h \<turnstile> (C, fs) \<surd>" shows "P,h \<turnstile> (C, fs((F, D) \<mapsto> v)) \<surd>"
card_of_Times_infinite_simps_2: fixes A :: "'a set"   and B :: "'b set" assumes "infinite A"   and "B \<noteq> {}"   and "|B| \<le>o |A|" shows "|A| =o |A \<times> B|"
sep_impl_def: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "P \<longrightarrow>\<^sup>* Q \<equiv> \<lambda>h. \<forall>h' h''. h \<oplus> h' = Some h'' \<and> P h' \<longrightarrow> Q h''"
elem_exact: fixes S :: "('c, 'd) eq_object_scheme"   and a :: "'c"   and A :: "'c set" assumes "equivalence S"   and "a \<in> carrier S"   and "a \<in> A" shows "a .\<in>\<^bsub>S\<^esub> A"
elim_substitute': fixes f :: "'b \<Rightarrow> 'b ttree"   and T :: "'b set"   and t :: "'b ttree"   and xs :: "'b list" shows "substitute' f T t xs = (xs \<in> paths (substitute f T t))"
bisimSubstExtBang: fixes P :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b" assumes "guarded P" shows "\<Psi> \<rhd> !P \<sim>\<^sub>s P \<parallel> !P"
get_spec_def: fixes M :: "('b SecurityInvariant \<times> 'b SecurityInvariant_configured) list" shows "get_spec M \<equiv> map snd M"
hyp_raw: fixes c2 :: "('a, 'b, 'c) com"   and s2 :: "('a, 'c) xstate"   and y :: "('a, 'b, 'c) com \<times> ('a, 'c) xstate" assumes "\<Gamma>\<turnstile> (c1, s1) \<rightarrow> (c2, s2)"   and "\<Gamma>\<turnstile> (c, s) \<rightarrow>\<^sup>* (c2, s2)"   and "y = (c2, s2)" shows "\<Gamma>\<turnstile>c2 \<down> s2"
distinct_keys: fixes t :: "('a, 'b) RBT.rbt" shows "distinct (keys t)"
pickInp_def: fixes inp :: "'e \<Rightarrow> 'f set option" shows "pickInp inp \<equiv> lift pick inp"
execute_def: fixes name :: "char list"   and s :: "nat \<Rightarrow> nat option" shows "execute name s \<equiv> case lookup_operator name of Some (x, xa, effs, xb) \<Rightarrow> s ++ map_of (map (\<lambda>(uu_, x, uu_, v). (x, v)) (filter (eff_enabled s) effs))"
map_add_split: fixes f :: "'c \<Rightarrow> 'd option"   and A :: "'c set"   and B :: "'c set" assumes "dom f = A \<union> B" shows "f |` A ++ f |` B = f"
f_meas: shows "f \<in> M \<rightarrow>\<^sub>M N"
boundedI: fixes b :: "'a word"   and a :: "'a word"   and c :: "'a word" assumes "b \<le>s a"   and "c \<le>s a" shows "signed.max b c \<le>s a"
inf2: fixes B :: "'b"   and k :: "'b"   and b :: "'b"   and a :: "'b" assumes "B \<equiv> k \<sqinter> b" shows "a \<sqinter> B \<equiv> k \<sqinter> (a \<sqinter> b)"
IH: assumes "n \<le> e_length b" shows "eval r_update_aux [n, b, j, v] \<down>= Partial_Recursive.list_encode ((take n (Partial_Recursive.list_decode b))[j := v])"
star_inductr_var_eq2: fixes y :: "'a"   and x :: "'a" assumes "y ; x = y" shows "y ; x\<^sup>\<star> = y"
ldropWhile_LNil: fixes P :: "'c \<Rightarrow> bool" shows "ldropWhile P LNil = LNil"
vars_FnCall: fixes F :: "nat"   and Xs :: "exp list" shows "vars (FnCall F Xs) = vars_list Xs"
simps_2: fixes l :: "'b tree234"   and uu :: "'b"   and r :: "'b tree234" shows "bal \<langle>l, uu, r\<rangle> = (bal l \<and> bal r \<and> height l = height r)"
prod_norm: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" shows "(\<Prod>x\<in>A. norm (f x)) = norm (prod f A)"
wf_interp_for_formula_any_Inl: fixes w :: "'a list"   and p :: "nat"   and I :: "(nat + nat set) list"   and \<phi> :: "'a formula" assumes "wf_interp_for_formula (w, Inl p # I) \<phi>" shows "\<forall>p<length w. wf_interp_for_formula (w, Inl p # I) \<phi>"
f_density: fixes i :: "nat" shows "density (density M (indicator (Q i))) (f i) = density N (indicator (Q i))"
exists_cong_per: fixes A :: "'p"   and B :: "'p"   and X :: "'p"   and Y :: "'p" shows "\<exists>C. Per A B C \<and> Cong B C X Y"
degree_m_eq_monic: fixes f :: "int poly"   and m :: "int" assumes "monic f"   and "1 < m" shows "degree (poly_mod.Mp m f) = degree f"
n_rw: shows "nrows_iarray (IArray (map (\<lambda>x. IArray (map (\<lambda>i. A $ mod_type_class.from_nat x $ mod_type_class.from_nat i) [0..<CARD('b)])) [0..<CARD('c)])) = CARD('c)"
thread_oks_bisim_inv: fixes ts1 :: "'t \<Rightarrow> ('a \<times> 'b \<Rightarrow>f nat) option"   and ts2 :: "'t \<Rightarrow> ('c \<times> 'd \<Rightarrow>f nat) option"   and tas1 :: "('t, 'x1, 'm1) new_thread_action list"   and tas2 :: "('t, 'x2, 'm2) new_thread_action list" assumes "\<forall>t. (ts1 t = None) = (ts2 t = None)"   and "list_all2 (nta_bisim bisim) tas1 tas2" shows "thread_oks ts1 tas1 = thread_oks ts2 tas2"
exec_stop_vs_control_independence': fixes upd :: "('b \<Rightarrow> 'b) \<Rightarrow> 'c control_state_scheme \<Rightarrow> 'c control_state_scheme"   and f :: "'b \<Rightarrow> 'b"   and b :: "bool"   and \<sigma> :: "'c control_state_scheme" assumes "\<sharp> upd" shows "upd f (\<sigma>\<lparr>return_status := b\<rparr>) = upd f \<sigma>\<lparr>return_status := b\<rparr>"
total_on_def: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" shows "Restricted_Predicates.total_on P A = (\<forall>x\<in>A. \<forall>y\<in>A. x = y \<or> P x y \<or> P y x)"
n_ge: fixes i :: "nat" assumes "1 \<le> i"   and "i < k" shows "n (Suc i) \<le> n i"
assms: shows "child p (Grid_Point.inv dir) d = child ps dir d"
hyps_2: shows "P \<sim>\<lbrakk>\<cdot>\<rbrakk><SRel,TRel> Q"
assms_2: fixes x :: "'a" assumes "x \<in> A" shows "countable (B x)"
ec: shows "spr_simAbs ec = SPRsingle.sim_equiv_class agent t"
distinct_in_set_tlD: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set (tl xs)"   and "distinct xs" shows "x \<noteq> hd xs"
absorb_iff2: fixes b :: "'d upred"   and a :: "'d upred" shows "(a \<sqsubseteq> b) = ((a \<and> b) = b)"
Q: shows "Q ** fst (foldl (echelon_form_of_column_k bezout) (A, 0) [0..<Suc k]) = fst (echelon_form_of_column_k bezout (fst (foldl (echelon_form_of_column_k bezout) (A, 0) [0..<Suc k]), snd (foldl (echelon_form_of_column_k bezout) (A, 0) [0..<Suc k])) (Suc k))"
\<alpha>_def: fixes G :: "nat set \<times> nat set set" shows "\<alpha> G \<equiv> SUP vs\<in>independent_sets G. enat (card vs)"
a: shows "\<forall>a\<in>A. (\<lambda>a\<in>A. if a \<in> B then f (generate G {a}) else \<one>) a = \<one>"
AboveS_def: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" shows "AboveS r A \<equiv> {b \<in> Field r. \<forall>a\<in>A. b \<noteq> a \<and> (a, b) \<in> r}"
measure_countable_Union_le: fixes S :: "nat \<Rightarrow> 'a set"   and M :: "'a measure"   and B :: "real" assumes "\<And>i. S i \<in> fmeasurable M"   and "\<And>n. Sigma_Algebra.measure M (\<Union> (S ` {..n})) \<le> B" shows "Sigma_Algebra.measure M (\<Union> (range S)) \<le> B"
from_hma\<^sub>m_def: fixes a :: "(('e, 'f) Finite_Cartesian_Product.vec, 'g) Finite_Cartesian_Product.vec" shows "from_hma\<^sub>m a = Matrix.mat CARD('g) CARD('f) (\<lambda>(i, j). a $h Bij_Nat.from_nat i $h Bij_Nat.from_nat j)"
in_PLS_charles_leq_2_pow_n: fixes PROB :: "(('c, bool) fmap \<times> ('c, bool) fmap) set"   and s :: "('c, bool) fmap"   and as :: "(('c, bool) fmap \<times> ('c, bool) fmap) list" assumes "finite PROB"   and "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB" shows "\<exists>x. x \<in> PLS_charles s as PROB \<and> x \<le> 2 ^ card (prob_dom PROB) - 1"
echelon_form_of_column_k_det_iarrays_def: fixes A' :: "'c \<times> 'c iarray iarray \<times> nat \<times> ('c \<Rightarrow> 'c \<Rightarrow> 'c \<times> 'c \<times> 'c \<times> 'c \<times> 'c)"   and k :: "nat" shows "echelon_form_of_column_k_det_iarrays A' k = (let (det_P, A, i, bezout) = A' in if i \<noteq> nrows_iarray A \<and> A !! i !! k = (0::'c) \<and> \<not> vector_all_zero_from_index (i + 1, column_iarray k A) then (- (1::'c) * det_P, echelon_form_of_column_k_iarrays (A, i, bezout) k) else (det_P, echelon_form_of_column_k_iarrays (A, i, bezout) k))"
IH: assumes "\<And>x. x \<in> A \<Longrightarrow> finite (B x)"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> B x\<rbrakk> \<Longrightarrow> g x y \<in> carrier G" shows "(\<Otimes>x\<in>A. finprod G (g x) (B x)) = (\<Otimes>a\<in>Sigma A B. case a of (a, b) \<Rightarrow> g a b)"
Suc: fixes \<sigma> :: "'state" shows "small_nstep P \<sigma> (Suc n) = {\<sigma>2. \<exists>\<sigma>1. \<sigma>1 \<in> small P \<sigma> \<and> \<sigma>2 \<in> small_nstep P \<sigma>1 n}"
pp_arc_linorder_matrix_2: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" assumes "\<exists>e. f e \<noteq> bot \<and> (\<forall>d. e \<noteq> d \<longrightarrow> f d = bot)" shows "matrix_stone_relation_algebra.pp_arc f"
cnorm_eq_square: fixes x :: "'a"   and a :: "real" shows "(norm x = a) = (0 \<le> a \<and> x \<bullet>\<^sub>C x = complex_of_real (a\<^sup>2))"
Release_ltln_2: fixes w :: "nat \<Rightarrow> 'a set" shows "w \<Turnstile>\<^sub>n rewrite_syn_imp \<psi> = w \<Turnstile>\<^sub>n \<psi>"
find_indices_empty: fixes x :: "'c"   and Vs :: "'c list" shows "(find_indices x Vs = []) = (x \<notin> set Vs)"
pp_init_f_preflow: shows "NPreflow c s t pp_init_f"
monoid_axioms: shows "monoid G (\<cdot>) \<one>"
vsv: fixes a :: "V"   and b :: "V"   and c :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> f"   and "\<langle>a, c\<rangle> \<in>\<^sub>\<circ> f" shows "b = c"
insertI2: fixes a :: "'b"   and B :: "'b set"   and b :: "'b" assumes "a \<in> B" shows "a \<in> insert b B"
subst_Var: fixes i :: "nat"   and s :: "db"   and k :: "nat" shows "DBVar i<s/k> = (if k < i then DBVar (i - 1) else if i = k then s else DBVar i)"
NE_def: shows "NE = (\<lambda>x w. \<forall>xa. (xa ess x m\<rightarrow> \<box> (\<exists> xa)) w)"
Struct_axioms: shows "GE.SStruct eintT eintF eintP"
simps_3: fixes x :: "'b"   and xt :: "'b bintree"   and y :: "'b"   and yl :: "'b bintree"   and yr :: "'b bintree"   and ts :: "'b bintree list" shows "t_ins_aux \<lparr>t_type.folding = True, item = x, subtrees = xt # Branch y yl yr # ts\<rparr> = (if x \<le> y then t_ins_aux \<lparr>t_type.folding = True, item = x, subtrees = Branch y xt yr # ts\<rparr> else t_ins_aux \<lparr>t_type.folding = True, item = x, subtrees = Branch y yl xt # ts\<rparr>)"
less_eq_zmultiset_def: fixes M' :: "'b zmultiset"   and M :: "'b zmultiset" shows "(M' \<le> M) = (M' < M \<or> M' = M)"
p_g_is_p_g: shows "(\<lambda>g. Mapping.lookup (Mapping (step.p_g g))) = step.p_g"
\<mu>'_def: fixes i :: "nat"   and j :: "nat" shows "\<mu>' i j \<equiv> d (Suc j) * \<mu> i j"
elims: fixes x :: "status"   and xa :: "status"   and y :: "status" assumes "merge_status x xa = y" obtains uu_ :: "status" where "x = FAILED" and "xa = uu_" and "y = FAILED"   | "x = SUCCESS" and "xa = FAILED" and "y = FAILED"   | "x = FOUND" and "xa = FAILED" and "y = FAILED"   | "x = FOUND" and "xa = SUCCESS" and "y = FOUND"   | "x = FOUND" and "xa = FOUND" and "y = FOUND"   | "x = SUCCESS" and "xa = FOUND" and "y = FOUND"   | "x = SUCCESS" and "xa = SUCCESS" and "y = SUCCESS"
eq_neg_iff_add_eq_0: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.group_add minus plus zero uminus" shows "(a = uminus b) = (plus a b = zero)"
set_ext_strong_lift: fixes R :: "'alt \<Rightarrow> 'alt \<Rightarrow> bool"   and A :: "'alt set"   and B :: "'alt set" assumes "total_preorder_on alts' R"   and "A \<noteq> {}"   and "B \<noteq> {}"   and "A \<subseteq> alts'"   and "B \<subseteq> alts'"   and "A \<prec>[set_ext R] B" shows "A \<prec>[set_ext (lift_prefs alts' alts R)] B"
fbox_g_orbital_guard: fixes H :: "'b \<Rightarrow> bool"   and G :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and U :: "'b \<Rightarrow> real set"   and S :: "'b set"   and t\<^sub>0 :: "real" assumes "H = (\<lambda>s. G s \<and> Q s)" shows "|x\<acute>=f & G on U S @ t\<^sub>0] Q = |x\<acute>=f & G on U S @ t\<^sub>0] H"
lm014: fixes X :: "'c set" shows "all_partitions X = all_partitions' X"
nth_ucast: fixes w :: "'d word"   and n :: "nat" shows "ucast w !! n = (w !! n \<and> n < LENGTH('c))"
UP_root_rem_def: fixes R :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'c"   and a :: "'c" assumes "UP_cring R" shows "UP_cring.UP_root_rem R f a = monom (UP R) (UP_cring.taylor R a f 0) 0"
triangle_in_hom_4: shows "\<guillemotleft>\<r>[f\<^sup>*] \<cdot> (f\<^sup>* \<star> \<rho>.\<epsilon>) \<cdot> \<a>[f\<^sup>*, f, f\<^sup>*] \<cdot> (\<rho>.\<eta> \<star> f\<^sup>*) \<cdot> \<l>\<^sup>-\<^sup>1[f\<^sup>*] : f\<^sup>* \<Rightarrow> f\<^sup>*\<guillemotright>"
append_eq_append_conv_if: fixes xs\<^sub>1 :: "'a list"   and xs\<^sub>2 :: "'a list"   and ys\<^sub>1 :: "'a list"   and ys\<^sub>2 :: "'a list" shows "(xs\<^sub>1 @ xs\<^sub>2 = ys\<^sub>1 @ ys\<^sub>2) = (if length xs\<^sub>1 \<le> length ys\<^sub>1 then xs\<^sub>1 = take (length xs\<^sub>1) ys\<^sub>1 \<and> xs\<^sub>2 = drop (length xs\<^sub>1) ys\<^sub>1 @ ys\<^sub>2 else take (length ys\<^sub>1) xs\<^sub>1 = ys\<^sub>1 \<and> drop (length ys\<^sub>1) xs\<^sub>1 @ xs\<^sub>2 = ys\<^sub>2)"
leg1_simps_3: shows "src (g \<star> w) = src (src w)"
Sup_llist_def: fixes Xs :: "'a set llist" shows "Sup_llist Xs = \<Union> (lnth Xs ` {i. enat i < llength Xs})"
assms_4: shows "f' \<in> A \<rightarrow>\<^sub>E B"
add_le_left: fixes \<alpha> :: "V"   and \<beta> :: "V" assumes "Ord \<alpha>"   and "Ord \<beta>" shows "\<beta> \<subseteq>\<^sub>\<circ> \<alpha> + \<beta>"
select_convs_2: fixes verts :: "'d set"   and arcs :: "'c set"   and tail :: "'c \<Rightarrow> 'd"   and head :: "'c \<Rightarrow> 'd"   and more :: "'e" shows "arcs \<lparr>verts = verts, arcs = arcs, tail = tail, head = head, \<dots> = more\<rparr> = arcs"