hcju/isabelleps huggingface dataset:
axis_def: fixes k :: "'c"   and x :: "'b" shows "axis k x = (\<chi>i. if i = k then x else (0::'b))"
prod_dfs_bij: fixes A :: "nat bdd list \<times> bool list"   and B :: "nat bdd list \<times> bool list"   and n :: "nat"   and x :: "nat \<times> nat"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "prod_DFS A B n"   and "prod_is_node A B x" shows "(fst (prod_dfs A B x) ! i ! j = Some k \<and> dfa_is_node A i \<and> dfa_is_node B j) = (k < length (snd (prod_dfs A B x)) \<and> snd (prod_dfs A B x) ! k = (i, j))"
elims: fixes x :: "'a list"   and xa :: "'a \<Rightarrow> nat"   and xb :: "nat"   and xc :: "nat"   and xd :: "nat"   and y :: "nat" assumes "lsteps x xa xb xc xd = y" obtains f :: "'a \<Rightarrow> nat" and l :: "nat" and up :: "nat" and left :: "nat" where "x = []" and "xa = f__" and "xb = l__" and "xc = up__" and "xd = left__" and "y = 0"   | xe :: "'a" and xs :: "'a list" and f :: "'a \<Rightarrow> nat" and l :: "nat" and up :: "nat" and left :: "nat" where "x = xe # xs" and "xa = f" and "xb = l" and "xc = up" and "xd = left" and "y = (if f xe < l then lsteps xs f l up left else if l < f xe then up + lsteps (xe # xs) f (l + 1) up left else left + lsteps xs f l up left)"
step_1_most_general: fixes outcomes :: "'a set"   and \<R> :: "('a pmf \<times> 'a pmf) set"   and x :: "'a pmf"   and y :: "'a pmf"   and \<alpha> :: "real"   and \<beta> :: "real" assumes "rational_preference (\<P> outcomes) \<R>"   and "independent_vnm (\<P> outcomes) \<R>"   and "x \<succeq>[\<R>] y \<and> (y, x) \<notin> \<R>"   and "\<alpha> \<in> {0..1}"   and "\<beta> \<in> {0..1}"   and "\<beta> < \<alpha>" shows "mix_pmf \<alpha> x y \<succeq>[\<R>] mix_pmf \<beta> x y \<and> (mix_pmf \<beta> x y, mix_pmf \<alpha> x y) \<notin> \<R>"
fundamental_lemma_bad: fixes bad1 :: "'a \<Rightarrow> bool"   and bad2 :: "'b \<Rightarrow> bool"   and A :: "'a \<Rightarrow> bool"   and B :: "'b \<Rightarrow> bool"   and p :: "'a spmf"   and q :: "'b spmf" assumes "rel_spmf (\<lambda>x y. bad1 x = bad2 y \<and> (\<not> bad2 y \<longrightarrow> A x = B y)) p q" shows "Sigma_Algebra.measure (measure_spmf p) {x. bad1 x} = Sigma_Algebra.measure (measure_spmf q) {y. bad2 y}"
rk_acAC\<beta>: fixes R :: "Points"   and A :: "Points"   and a :: "Points"   and C :: "Points"   and c :: "Points"   and \<beta> :: "Points"   and Q :: "Points"   and A' :: "Points" assumes "rk {R, A, a} = 2"   and "rk {R, C, c} = 2"   and "rk {A, C} = 2"   and "rk {A, C, \<beta>} = 2"   and "rk {Q, A', a} = 2"   and "4 \<le> rk {A, A', C, a}" shows "rk {a, c, A, C, \<beta>} = 3"
diff_diff_ennreal': fixes z :: "ennreal"   and y :: "ennreal"   and x :: "ennreal" assumes "z \<le> y"   and "y - z \<le> x" shows "x - (y - z) = x + z - y"
strip_eq_While: fixes C :: "'a acom"   and b :: "bexp"   and c1 :: "com" shows "(strip C = WHILE b DO c1) = (\<exists>I P C1 Q. C = {I} WHILE b DO {P} C1 {Q} \<and> strip C1 = c1)"
\<theta>_conv_\<MM>_integral: fixes x :: "real" assumes "2 \<le> x" shows "(\<MM> has_integral \<MM> x * x - \<theta> x) {2..x}"
not_gr_zero: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "'c" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "(\<not> less zero n) = (n = zero)"
iptables_bigstep_add_match_notnot_simp: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and m :: "'c match_expr"   and rs :: "'c rule list"   and s :: "state"   and t :: "state" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>add_match (MatchNot (MatchNot m)) rs, s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile> \<langle>add_match m rs, s\<rangle> \<Rightarrow> t"
P1: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "|~ \<box>F \<longrightarrow> \<circle>F"
select_convs_3: fixes lenf :: "nat"   and front :: "'b list"   and status :: "'b status"   and rear :: "'b list"   and lenr :: "nat"   and more :: "'c" shows "status \<lparr>lenf = lenf, front = front, status = status, rear = rear, lenr = lenr, \<dots> = more\<rparr> = status"
assoc_2: fixes shows_prec :: "nat \<Rightarrow> 'a \<Rightarrow> char list \<Rightarrow> char list"   and shows_list :: "'a list \<Rightarrow> char list \<Rightarrow> char list"   and xs :: "'a list"   and r :: "char list"   and s :: "char list" assumes "class.show shows_prec shows_list" shows "shows_list xs r @ s = shows_list xs (r @ s)"
mult_onel: fixes x :: "'a" shows "x \<cdot> (1::'a) = x"
adm: fixes f :: "'f"   and g :: "'f"   and n :: "nat" assumes "w (f, 1) = 0" shows "pr_weak' (f, 1) (g, n)"
case_eq_if: fixes f :: "'b \<Rightarrow> 'd \<Rightarrow> 'a"   and prod :: "'b \<times> 'd" shows "(case prod of (x, xa) \<Rightarrow> f x xa) = f (fst prod) (snd prod)"
square_less_square: fixes x :: "'a"   and y :: "'a" assumes "(0::'a) \<le> x"   and "(0::'a) \<le> y" shows "(x * x < y * y) = (x < y)"
intro: fixes m :: "int"   and f :: "'b mod_ring poly" assumes "poly_mod_type TYPE('b) m"   and "poly_mod_type_irr_axioms f" shows "poly_mod_type_irr m f"
fUnionM_empty: shows "fUnionM {||} = return {||}"
ftf_syn: fixes u :: "'c event list"   and t :: "'c event list"   and s :: "'c event list"   and A :: "'c set" assumes "front_tickFree u"   and "front_tickFree t"   and "s setinterleaves ((t, u), insert tick (ev ` A))" shows "front_tickFree s"
pOp_assocTr43: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a \<in> carrier A"   and "b \<in> carrier A"   and "c \<in> carrier A"   and "d \<in> carrier A" shows "a \<plusminus> b \<plusminus> (c \<plusminus> d) = a \<plusminus> (b \<plusminus> c) \<plusminus> d"
lex_append_leftD: fixes r :: "('g \<times> 'g) set"   and xs :: "'g list"   and ys :: "'g list"   and zs :: "'g list" assumes "\<forall>x. (x, x) \<notin> r"   and "(xs @ ys, xs @ zs) \<in> lex r" shows "(ys, zs) \<in> lex r"
Max_wrt_among_total_preorder: fixes A :: "'alt set" shows "Max_wrt_among Ri' A = {x \<in> alts \<inter> A. \<forall>y\<in>alts \<inter> A. Ri' y x}"
indep_region_coregion: shows "\<V> \<bowtie> \<C>"
zero_order_2: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "'c" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less_eq n zero = (n = zero)"
simps_2: fixes h :: "'b \<times> bool"   and t :: "('b \<times> bool) list"   and l :: "nat list" shows "levelsCounter_aux (h # t) l = (if marked h then levelsCounter_aux t (l @ [1]) else levelsCounter_aux t (butlast l @ [Suc (last l)]))"
RH': shows "relH {a. a < lim h \<and> a \<notin> as} h (the_state \<sigma>')"
assms_2: fixes i :: "V" assumes "i \<in>\<^sub>\<circ> I" shows "\<phi> i : \<CC> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<AA> i"
invar_insert: fixes t :: "'b trie_map"   and xs :: "'b list" assumes "invar t" shows "invar (Trie_Map.insert xs t)"
add_zero2: fixes x :: "'a" shows "(0::'a) \<oplus> x = Some x"
countable_type_countable: fixes t :: "pdf_type" assumes "countable_type t" shows "countable (space (stock_measure t))"
finalsE: fixes es :: "'a exp list" assumes "finals es" obtains vs :: "val list" where "es = map Val vsa__"   | vs :: "val list" and a :: "nat" and es' :: "'a exp list" where "es = map Val vs @ Throw a # es'"
space_lim: shows "space local.lim = space (Pi\<^sub>M UNIV (\<lambda>_. borel))"
Sigma_msetE: fixes c :: "'b \<times> 'c"   and A :: "'b multiset"   and B :: "'b \<Rightarrow> 'c multiset" assumes "c \<in># Sigma_mset A B" obtains x :: "'b" and y :: "'c" where "x \<in># A" and "y \<in># B x" and "c = (x, y)"
exhaust: fixes y :: "Dom" obtains "y = low"   | "y = high"
hyps_3: assumes "\<And>x. x \<in> A \<Longrightarrow> \<exists>y. y \<in> A \<and> R x y" shows "A = {}"
inverse_arrows_compose: fixes g :: "'s"   and f :: "'s"   and f' :: "'s"   and g' :: "'s" assumes "S.seq g f"   and "S.inverse_arrows f f'"   and "S.inverse_arrows g g'" shows "S.inverse_arrows (g \<cdot>\<^sub>S f) (f' \<cdot>\<^sub>S g')"
exec_iff_exec_merge_guards: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t = \<Gamma>\<turnstile> \<langle>merge_guards c,s\<rangle> \<Rightarrow> t"
expand: fixes converter :: "('a, 'b, 'c, 'd) converter"   and converter' :: "('a, 'b, 'c, 'd) converter" assumes "run_converter converter = run_converter converter'" shows "converter = converter'"
hyps_1: fixes p :: "'b list"   and v :: "'a" assumes "n = length p"   and "awalk s p v" shows "dist v \<le> ereal (awalk_cost c p)"
prv_all_imp_cnj: fixes x :: "'var"   and \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla"   and "bprv (all x (imp \<phi> (imp \<psi> \<chi>)))" shows "bprv (all x (imp (cnj \<psi> \<phi>) \<chi>))"
hyps_3: fixes d :: "nat" assumes "X \<noteq> {}" shows "card (deg_sect X d) = d + (card X - 1) choose (card X - 1)"
simps_3: fixes v :: "game" shows "Composeo undefg (Agame v) = undefg"
enat_the_enat_\<delta>_times_arity_sym: fixes f :: "'s" shows "enat (the_enat (enat \<delta> * arity_sym f)) = enat \<delta> * arity_sym f"
elims_3: fixes x :: "('a \<times> constraint) list"   and xa :: "'a set"   and xb :: "'a simplex_state" assumes "\<not> invariant_simplex x xa xb" obtains cs :: "('a \<times> constraint) list" and J :: "'a set" and 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" where "x = cs" and "xa = J" and "xb = Simplex_State s" and "\<not> invariant_simplex' cs J s"
valid_prefixes_def: fixes r :: "('c, 'd) routing_rule_scheme list" shows "valid_prefixes r = foldr (\<and>) (map (\<lambda>rr. valid_prefix (routing_match rr)) r) True"
1: fixes s :: "char list \<Rightarrow> int" shows "pre C Q' s \<le> pre C Q s"
Cons_2: fixes lb :: "real" assumes "\<forall>p\<in>set (p # ps). \<forall>x. poly p x = 0 \<longrightarrow> lb < x"   and "\<forall>x\<le>lb. \<forall>p\<in>set (p # ps). sgn (poly p x) = sgn_neg_inf p"   and "lb < b" shows "thesis"
dsumag_aGroup: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)" shows "aGroup (a\<Oplus>\<^bsub>I\<^esub> A)"
Ay: shows "y n \<in> carrier (residue_ring (p ^ n))"
cfE_of_ss_invE: fixes cf :: "nat \<times> nat \<Rightarrow> 'capacity" shows "cfE_of cf \<subseteq> E \<union> E\<inverse>"
one_closed: fixes H :: "'a set"   and G :: "('a, 'b) monoid_scheme" assumes "subgroup H G" shows "\<one>\<^bsub>G\<^esub> \<in> H"
comm_ring_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "comm_ring_hom hom \<equiv> ring_hom hom"
r_find_k_2: fixes b :: "nat" shows "let k = the (eval r_find_k [b]) in \<phi> (the (s (e_append_zeros b k))) (e_length b + k) \<down>= 0"
xeq_def: fixes x :: "Value"   and s :: "Store"   and t :: "Store" shows "s \<equiv>[x] t = (alive x s = alive x t \<and> (\<forall>l. s\<turnstile> l reachable_from x \<longrightarrow> s@@l = t@@l))"
excess_eq: fixes t :: "nat"   and q :: "nat" assumes "t + q \<le> 7" shows "excessAtType t q 0 + t * \<d> 3 + q * \<d> 4 = \<b> t q"
deleteAroundCons: fixes g :: "graph"   and a :: "nat"   and p :: "nat \<times> nat"   and ps :: "(nat \<times> nat) list" shows "deleteAround g a (p # ps) = (if fst p \<in> {v. \<exists>f\<in>set (facesAt g a). |vertices f| = 4 \<and> v \<in> {f \<bullet> a, f \<bullet> (f \<bullet> a)} \<or> |vertices f| \<noteq> 4 \<and> v = f \<bullet> a} then deleteAround g a ps else p # deleteAround g a ps)"
residualInject_1: fixes x3 :: "subject"   and x1 :: "name"   and x2 :: "pi"   and y3 :: "subject"   and y1 :: "name"   and y2 :: "pi" shows "(x3\<guillemotleft>x1\<guillemotright> \<prec> x2 = y3\<guillemotleft>y1\<guillemotright> \<prec> y2) = (x3 = y3 \<and> [x1].x2 = [y1].y2)"
ih: fixes ta :: "('s, 'v) tm"   and sa :: "('s, 'v) tm" assumes "hsize ta + hsize sa < hsize t + hsize s"   and "ta >\<^sub>t sa" shows "vars sa \<subseteq> vars ta"
2: fixes tcs :: "(String.literal \<times> (String.literal \<times> String.literal set list) list) list"   and n :: "String.literal"   and ar :: "nat" assumes "dom (map_of (map (apsnd map_of) tcs)) = dom (map_of tao)"   and "alist_conds tao"   and "(n, ar) \<in> set tao" shows "n \<in> fst ` set tcs"
strict_prefix_butlast: fixes xs :: "'b list"   and ys :: "'b list" assumes "prefix xs (butlast ys)"   and "ys \<noteq> []" shows "strict_prefix xs ys"
cases_scheme: fixes r :: "'b heap_scheme" obtains arrays :: "typerep \<Rightarrow> nat \<Rightarrow> nat list" and refs :: "typerep \<Rightarrow> nat \<Rightarrow> nat" and lim :: "nat" and more :: "'b" where "r = \<lparr>arrays = arrays, refs = refs, lim = lim, \<dots> = more\<rparr>"
list_comb_nesting: fixes f :: "'c"   and xs :: "'c list" shows "left_nesting (list_comb f xs) = left_nesting f + length xs"
insert_map_def: fixes i :: "nat"   and z :: "'c"   and f :: "nat \<Rightarrow> 'c"   and k :: "nat" shows "insert_map i z f k \<equiv> if k < i then f k else if k = i then z else f (k - 1)"
intro: fixes G :: "('c, 'd) monoid_scheme" assumes "\<And>a b. \<lbrakk>a \<in> carrier G; b \<in> carrier G\<rbrakk> \<Longrightarrow> \<exists>c. c \<in> carrier G \<and> c gcdof\<^bsub>G\<^esub> a b" shows "gcd_condition_monoid_axioms G"
icrit_spec_product_crit: shows "punit.icrit_spec punit.product_crit"
hyps: shows "replicate n (v 0) \<frown> v \<approx> v"
valid_lists'_valid_lists_aux: fixes S :: "Vars.var set"   and G :: "CoCalls"   and prefix :: "Vars.var set"   and xs :: "Vars.var list"   and x :: "Vars.var" assumes "valid_lists' S G prefix xs"   and "x \<in> prefix" shows "set xs G\<times> {x} \<sqsubseteq> G"
Max_wrt_among_nonempty: fixes B :: "'a set" assumes "B \<inter> carrier \<noteq> {}"   and "finite (B \<inter> carrier)" shows "Max_wrt_among S B \<noteq> {}"
euclidean_size_ratfps_def: fixes x :: "'a ratfps" shows "euclidean_size x = (if x = 0 then 0 else 2 ^ ratfps_subdegree x)"
inversion_number_between_def: fixes xs :: "'a list"   and ys :: "'a list" shows "inversion_number_between xs ys = card (inversions_between xs ys)"
f12_def: shows "f12 = fst (split_face f v u ws)"
assms_2: shows "asset \<in> support_set pf1 \<union> B"
sup_least: fixes A :: "'a set"   and z :: "'a" assumes "A \<subseteq> carrier L"   and "z \<in> carrier L"   and "\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq>\<^bsub>L\<^esub> z" shows "\<Squnion>\<^bsub>L\<^esub>A \<sqsubseteq>\<^bsub>L\<^esub> z"
s\<^sub>1s\<^sub>2: shows "(s\<^sub>1, s\<^sub>2) \<in> [\<leadsto>]\<^sup>="
IH_2: assumes "\<not> complete r" shows "2 ^ min_height r < size1 r"
iso_imp_comm_group: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme" assumes "comm_group G"   and "G \<cong> H"   and "Group.monoid H" shows "comm_group H"
c: shows "\<chi> \<in> K \<^bold>* (\<phi> .\<or>. \<psi>)"
HFail_HInv5: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HFail s s' p"   and "HInv5_inner s q"   and "Inv2a s'" shows "HInv5_inner s' q"
x2: fixes x :: "'a" assumes "x \<in> closure (projection (orthogonal_complement (space_as_set C)) ` space_as_set A)" shows "x \<in> space_as_set B"
elims_2: fixes x :: "'b heap" assumes "no_Empty x" obtains xa :: "'b" and hs :: "'b heap list" where "x = Hp xa hs" and "no_Emptys hs"
assms_2: shows "\<forall>i\<in>I. subgroup (S i) G"
fmap_distinct_1: fixes k :: "'c"   and v :: "'d"   and m :: "('c, 'd) fmap" shows "{$$} \<noteq> m(k $$:= v)"
Inf_fset_transfer: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "bi_unique A"   and "bi_total A" shows "rel_fun (rel_set (rel_set A)) (rel_set A) (\<lambda>A. if finite (\<Inter> A) then \<Inter> A else {}) (\<lambda>A. if finite (\<Inter> A) then \<Inter> A else {})"
nzrows_def: fixes A :: "'c mat" shows "nzrows A = filter (\<lambda>r. r \<noteq> 0\<^sub>v (dim_col A)) (rows A)"
odiff_le_odiff: fixes x :: "V"   and y :: "V"   and z :: "V" assumes "x \<unlhd> y" shows "odiff x z \<le> odiff y z"
rel_game_def: fixes \<A> :: "'a \<times> 'a \<Rightarrow> nat spmf" shows "chaum_ped_\<Sigma>_commit.rel_game \<A> = TRY G \<bind> (\<lambda>(h, w). \<A> h \<bind> (\<lambda>w'. return_spmf ((h, w') \<in> R))) ELSE return_spmf False"
improper_integral_at_top: fixes a :: "real"   and f :: "real \<Rightarrow> 'a" assumes "f absolutely_integrable_on {a..}" shows "((\<lambda>x. integral {a..x} f) \<longlongrightarrow> integral {a..} f) at_top"
max_cases: fixes a :: "'a"   and b :: "'a" obtains "max a b = a"   | "max a b = b"
simps_10: fixes v :: "'a formula"   and va :: "'a formula" shows "is_lit_plus (v \<^bold>\<or> va) = False"
first_countable_basis: fixes x :: "'a" shows "\<exists>A. (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
hyps: fixes child_node :: "(_) node_ptr"   and parent :: "(_) object_ptr" assumes "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child_node = child"   and "h \<turnstile> get_parent child_node \<rightarrow>\<^sub>r Some parent"   and "h \<turnstile> get_dom_component parent \<rightarrow>\<^sub>r c" shows "parent \<in> set c"
orient_lit_semantics_replacement: fixes I :: "'a trm \<Rightarrow> 'a trm \<Rightarrow> bool"   and l :: "'a literal"   and u :: "'a trm"   and v :: "'a trm"   and polarity :: "sign"   and \<theta> :: "('a \<times> 'a trm) list"   and \<sigma> :: "('a \<times> 'a trm) list"   and u' :: "'a trm" assumes "fo_interpretation I"   and "orient_lit_inst l u v polarity \<theta>"   and "validate_ground_lit I (subst_lit l \<sigma>)"   and "I (u \<lhd> \<sigma>) (u' \<lhd> \<sigma>)" shows "validate_ground_lit I (subst_lit (mk_lit polarity (Eq u' v)) \<sigma>)"
rbt_less_simps_1: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and k :: "'a" shows "ord.rbt_less less k Empty = True"
subspace_tangent_space: fixes k :: "enat"   and p :: "'c" shows "subspace (manifold_eucl.tangent_space k p)"
arev_dom: fixes a :: "'b" shows "(a \<in> arcs G) = (arev a \<noteq> a)"
iso_def: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set"   and f :: "'d \<Rightarrow> 'e" shows "iso r r' f \<equiv> embed r r' f \<and> bij_betw f (Field r) (Field r')"
uncurry0_def: fixes c :: "'c" shows "uncurry0 c \<equiv> \<lambda>_. c"
gauss_jordan_main_code: fixes A :: "'b mat"   and B :: "'b mat"   and i :: "nat"   and j :: "nat" shows "gauss_jordan_main A B i j = (let nr = dim_row A; nc = dim_col A in if i < nr \<and> j < nc then let aij = A $$ (i, j) in if aij = (0::'b) then case concat (map (\<lambda>i'. if A $$ (i', j) \<noteq> (0::'b) then [i'] else []) [Suc i..<nr]) of [] \<Rightarrow> gauss_jordan_main A B i (Suc j) | i' # x \<Rightarrow> gauss_jordan_main (swaprows i i' A) (swaprows i i' B) i j else if aij = (1::'b) then let v = \<lambda>i. A $$ (i, j) in gauss_jordan_main (eliminate_entries v A i j) (eliminate_entries v B i j) (Suc i) (Suc j) else let iaij = inverse_class.inverse aij; A' = multrow i iaij A; B' = multrow i iaij B; v = \<lambda>i. A' $$ (i, j) in gauss_jordan_main (eliminate_entries v A' i j) (eliminate_entries v B' i j) (Suc i) (Suc j) else (A, B))"
phiUses_exI: fixes m :: "'node"   and g :: "'g"   and n :: "'node"   and v :: "'val"   and vs :: "'val list" assumes "m \<in> set (predecessors g n)"   and "phis g (n, v) = Some vs"   and "n \<in> set (\<alpha>n g)" obtains v' :: "'val" where "v' \<in> phiUses g m" and "v' \<in> set vs"
sdrop_shift: fixes i :: "nat"   and w :: "'d list"   and s :: "'d stream" shows "sdrop i (w @- s) = drop i w @- sdrop (i - length w) s"
assms: shows "s \<in> timpl_closure_set (\<Union> {M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<delta> |\<delta>. P \<delta>}) T"
cTf: fixes i :: "'b" assumes "i \<in> I" shows "closedin X (T i \<inter> f i -` U)"
simps_1: fixes x1 :: "'b form multiset"   and x2 :: "'b form multiset"   and y1 :: "'b form multiset"   and y2 :: "'b form multiset" shows "(( x1 \<Rightarrow>* x2) = ( y1 \<Rightarrow>* y2)) = (x1 = y1 \<and> x2 = y2)"
exhaust_sel: fixes name :: "name" obtains "name = Name (as_string name)"
path_conforms_with_strategy_1: shows "path_conforms_with_strategy p P \<sigma>"
intro: fixes return :: "'d \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e"   and altc :: "'f cset \<Rightarrow> ('f \<Rightarrow> 'e) \<Rightarrow> 'e" assumes "monad_altc return bind altc"   and "three TYPE('f)" shows "monad_altc3 return bind altc"
IH: fixes k :: "nat" shows "list.set (acc_lengths k as) \<subseteq> list.set (acc_lengths k as2)"
Abs_a: fixes b :: "'a" assumes "b \<in> Collect (Domainp A)" shows "A b (Abs b)"
sint_range': fixes x :: "'b word" shows "- (2 ^ (LENGTH('b) - Suc 0)) \<le> sint x \<and> sint x < 2 ^ (LENGTH('b) - Suc 0)"
indspaceD_transform: fixes f :: "('f, 'g) aezfun \<Rightarrow> 'v"   and G :: "'g set"   and FH :: "('f, 'g) aezfun set"   and V :: "'v set"   and g :: "'g"   and x :: "('f, 'g) aezfun"   and a :: "'f" assumes "f \<in> indspace G FH V"   and "g \<in> G"   and "x \<in> FH" shows "f (a \<delta>\<delta> g * x) = (a \<delta>\<delta> g) \<cdot> f x"
sympE: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and b :: "'e"   and a :: "'e" assumes "symp r"   and "r b a" obtains "r a b"
prio_emb_pos_def: fixes prio :: "dir list \<Rightarrow> nat"   and t :: "('c, 'd) tm"   and s :: "('c, 'd) tm" shows "prio_emb_pos prio t s = (ARG_MIN prio p. p \<noteq> [] \<and> position_of t p \<and> emb_step_at' p t \<unrhd>\<^sub>e\<^sub>m\<^sub>b s)"
preferred_alts_lift_pref_profile: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and agents' :: "'a set"   and alts' :: "'b set"   and i :: "'a"   and x :: "'b" assumes "pref_profile_wf agents alts R"   and "agents \<subseteq> agents'"   and "alts \<subseteq> alts'"   and "alts \<noteq> {}"   and "agents \<noteq> {}"   and "finite alts'"   and "i \<in> agents'"   and "x \<in> alts'" shows "preferred_alts (lift_pref_profile agents alts agents' alts' R i) x = (if i \<in> agents \<and> x \<in> alts then preferred_alts (R i) x else alts')"
assms: shows "\<forall>b\<in>set B. \<not> is_Insert b \<and> \<not> is_Delete b"
mv_box_correct': fixes m :: "nat"   and n :: "nat"   and lm :: "nat list" assumes "max m n < length lm"   and "m \<noteq> n" shows "\<exists>stp. abc_steps_l (0, lm) (mv_box m n) stp = (3, lm[n := lm ! m + lm ! n, m := 0])"
well_order_induct: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "\<And>x. \<forall>y. y \<noteq> x \<and> (y, x) \<in> r \<longrightarrow> P y \<Longrightarrow> P x" shows "P a"
split_sel_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b list \<Rightarrow> 'a"   and list :: "'b list" shows "P (case list of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = (\<not> (list = [] \<and> \<not> P f1 \<or> list = hd list # tl list \<and> \<not> P (f2 (hd list) (tl list))))"
count_rem_components_def: fixes bs :: "(('a \<Rightarrow>\<^sub>0 'd) \<times> 'e) list" shows "punit.count_rem_components bs = length (remdups (map local.punit.component_of_term (punit.Keys_to_list (map fst bs)))) - punit.count_const_lt_components (filter (\<lambda>b. fst b \<noteq> 0) bs)"
update_arg_wf_tuples': fixes n :: "nat"   and hops :: "nat"   and nhip :: "nat" assumes "Suc 0 \<le> n" shows "update_arg_wf (n, kno, val, hops, nhip)"
tr_sem_equiv: fixes A :: "('fun, 'var) stateful_strand_step list"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "fv\<^sub>s\<^sub>s\<^sub>t A \<inter> bvars\<^sub>s\<^sub>s\<^sub>t A = {}"   and "subst_domain \<I> = \<V> \<and> ground (subst_range \<I>)" shows "\<lbrakk>{}; {}; A\<rbrakk>\<^sub>s \<I> = (\<exists>A'\<in>set (tr A []). \<I> \<Turnstile> \<langle>A'\<rangle>)"
simps_2: fixes x :: "'a"   and xs :: "[:'a:]" assumes "x \<noteq> \<bottom>"   and "xs \<noteq> \<bottom>" shows "eq\<cdot>(x :# xs)\<cdot>[::] = FF"