hcju/isabelleps huggingface dataset:
filtermap_at_left_shift: fixes d :: "real"   and a :: "real" shows "filtermap (\<lambda>x. x - d) (at_left a) = at_left (a - d)"
0: shows "Cring_Poly.compose R p q = rev_compose R q p"
uint8_shiftl_def: fixes x :: "uint8"   and n :: "integer" shows "uint8_shiftl x n = (if n < 0 \<or> 8 \<le> n then undefined push_bit x n else push_bit (nat_of_integer n) x)"
prime_dvd_mult_eq_nat: fixes p :: "nat"   and m :: "nat"   and n :: "nat" assumes "prime p" shows "(p dvd m * n) = (p dvd m \<or> p dvd n)"
less_return_time_imp_exivl: fixes h :: "real"   and P :: "'a set"   and x' :: "'a" assumes "h \<le> return_time P x'"   and "returns_to P x'"   and "closed P"   and "0 \<le> h" shows "h \<in> existence_ivl0 x'"
freshResidualChain3_2: fixes xvec :: "name list"   and \<alpha> :: "'d action"   and P :: "('d, 'e, 'f) psi" assumes "xvec \<sharp>* (\<alpha> \<prec> P)"   and "xvec \<sharp>* bn \<alpha>" shows "xvec \<sharp>* P"
u_returns_ge: shows "\<forall>\<^sub>F y in at x. returns_to {x \<in> S. s x = 0} y \<and> return_time {x \<in> S. s x = 0} y \<le> u y"
elliptic_isometry_inv: fixes f :: "'b \<Rightarrow> 'b" assumes "elliptic_isometry f" shows "elliptic_isometry (inv f)"
nth_default_Cons_Suc: fixes dflt :: "'c"   and x :: "'c"   and xs :: "'c list"   and n :: "nat" shows "nth_default dflt (x # xs) (Suc n) = nth_default dflt xs n"
one_marker: shows "\<exists>!j. j < length (msgs (S t i) cid) \<and> msgs (S t i) cid ! j = Marker"
seqr_or_distl: fixes P :: "('d, 'f) urel"   and Q :: "('d, 'f) urel"   and R :: "('f, 'e) urel" shows "(P \<or> Q) ;; R = (P ;; R \<or> Q ;; R)"
bd_Cinfinite: shows "cinfinite bd_F1 \<and> Card_order bd_F1"
signed_take_bit_eq_if_positive: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and not :: "'c \<Rightarrow> 'c"   and a :: "'c"   and n :: "nat" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not"   and "\<not> bit a n" shows "ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not n a = take_bit n a"
NotNull: shows "\<not> (a = 0 \<and> b = 0 \<and> c = 0)"
step_3: shows "\<forall>p\<in>set (fst r). p \<in> derivable' R \<and> (\<exists>n. (p, n) \<in> derivable R)"
permutes_as_lists_succ: fixes xss :: "'b list list"   and f :: "'b \<Rightarrow> 'b" assumes "distincts xss"   and "\<And>xs. xs \<in> set xss \<Longrightarrow> list_succ xs = perm_restrict f (set xs)"   and "f permutes \<Union> (sset xss)" shows "f = lists_succ xss"
lt_uminus: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" shows "lt (- p) = lt p"
cases: fixes a1 :: "trm"   and a2 :: "trm" assumes "a1 \<longmapsto> a2" obtains v\<^sub>2 :: "trm" and T\<^sub>1\<^sub>1 :: "POPLmark.type" and t\<^sub>1\<^sub>2 :: "trm" where "a1 = (\<lambda>:T\<^sub>1\<^sub>1__. t\<^sub>1\<^sub>2__) \<bullet> v\<^sub>2" and "a2 = t\<^sub>1\<^sub>2__[0 \<mapsto> v\<^sub>2]" and "v\<^sub>2 \<in> value"   | T\<^sub>1\<^sub>1 :: "POPLmark.type" and t\<^sub>1\<^sub>2 :: "trm" and T\<^sub>2 :: "POPLmark.type" where "a1 = (\<lambda><:T\<^sub>1\<^sub>1. t\<^sub>1\<^sub>2) \<bullet>\<^sub>\<tau> T\<^sub>2" and "a2 = t\<^sub>1\<^sub>2[0 \<mapsto>\<^sub>\<tau> T\<^sub>2]"   | t :: "trm" and t' :: "trm" and u :: "trm" where "a1 = tb__ \<bullet> u" and "a2 = t'__ \<bullet> u" and "tb__ \<longmapsto> t'__"   | v :: "trm" and t :: "trm" and t' :: "trm" where "a1 = v \<bullet> tc__" and "a2 = v \<bullet> t'a__" and "v \<in> value" and "tc__ \<longmapsto> t'a__"   | t :: "trm" and t' :: "trm" and T :: "POPLmark.type" where "a1 = t \<bullet>\<^sub>\<tau> T" and "a2 = t' \<bullet>\<^sub>\<tau> T" and "t \<longmapsto> t'"
distrib_left: fixes a :: "'c \<times> 'c \<Rightarrow> 'd"   and b :: "'c \<times> 'c \<Rightarrow> 'd"   and c :: "'c \<times> 'c \<Rightarrow> 'd" shows "a \<odot> (b \<oplus> c) = a \<odot> b \<oplus> a \<odot> c"
strategyproofness: shows "pmf (sds R47) d + pmf (sds R47) a \<le> pmf (sds R4) d + pmf (sds R4) a"
Busemann_function_ereal: fixes xi :: "'b Gromov_completion"   and x :: "'b"   and y :: "'b" shows "ereal (Busemann_function_at xi x y) = Sup {limsup (\<lambda>n. ereal (dist x (u n) - dist y (u n))) |u. (\<lambda>n. to_Gromov_completion (u n)) \<longlonglongrightarrow> xi}"
Collection_def: fixes empty :: "'c"   and is_empty :: "'c \<Rightarrow> bool"   and of_list :: "'d list \<Rightarrow> 'c"   and multiset :: "'c \<Rightarrow> 'd multiset" shows "Collection empty is_empty of_list multiset \<equiv> ((\<forall>e. is_empty e \<longrightarrow> e = empty) \<and> is_empty empty) \<and> multiset empty = {#} \<and> (\<forall>i. multiset (of_list i) = mset i)"
lambdas3: fixes l :: "nat"   and vs :: "(nat \<times> char list) list"   and c :: "call"   and x :: "lambda" assumes "Lambda l vs c \<in> lambdas x" shows "set vs \<subseteq> vars x"
Ln'_times_of_real: fixes x :: "real"   and z :: "complex" assumes "0 < x"   and "z \<noteq> 0" shows "Ln' (complex_of_real x * z) = complex_of_real (ln x) + Ln' z"
AllN_6: shows "insertion (nth_default 0 ((drop z L)[var := B / C])) c = C"
ring_times_x_0: fixes R :: "('e, 'f) Ring_scheme"   and x :: "'e" assumes "Ring R"   and "x \<in> carrier R" shows "x \<cdot>\<^sub>r\<^bsub>R\<^esub> \<zero>\<^bsub>R\<^esub> = \<zero>\<^bsub>R\<^esub>"
IH: fixes y :: "nat"   and xs :: "'a list"   and t :: "'a tree"   and zs :: "'a list" assumes "y < n"   and "y \<le> length xs"   and "bal y xs = (t, zs)" shows "min_height t = nat \<lfloor>log 2 (real (y + 1))\<rfloor>"
alw_nxt: fixes \<phi> :: "'b stream \<Rightarrow> bool" shows "alw \<phi> = (\<lambda>xs. \<phi> xs \<and> nxt (alw \<phi>) xs)"
red_preserves_lconf: fixes e :: "char list exp"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "char list exp"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l' :: "char list \<Rightarrow> val option"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and E :: "char list \<Rightarrow> ty option"   and T :: "ty" assumes "P \<turnstile> \<langle>e,(h, l)\<rangle> \<rightarrow> \<langle>e',(h', l')\<rangle>"   and "P,E,h \<turnstile> e : T"   and "P,h \<turnstile> l (:\<le>) E" shows "P,h' \<turnstile> l' (:\<le>) E"
runit_simps_2: fixes f :: "'a" assumes "S.ide f" shows "S.src (S.runit f) = S.src f"
option_while_simps_4: fixes s :: "'d"   and ss' :: "'d option"   and C :: "'d \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'd option" assumes "(Some s, ss') \<in> option_while' C B" shows "option_while C B s = ss'"
MInfty_neq_PInfty_2: shows "- \<infinity> \<noteq> \<infinity>"
inv_permute: fixes p :: "perm" shows "inv (permute p) = permute (- p)"
preserves_hom: fixes f :: "'b"   and a :: "'b"   and b :: "'b" assumes "\<guillemotleft>f : a \<Rightarrow> b\<guillemotright>" shows "\<guillemotleft>\<rr>'.map f : local.map a \<Rightarrow> R b\<guillemotright>"
l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.type_wf ShadowRootClass.known_ptr ShadowRootClass.known_ptrs Shadow_DOM.get_parent Shadow_DOM.get_parent_locs Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs Shadow_DOM.get_ancestors Shadow_DOM.get_ancestors_locs Shadow_DOM.get_root_node Shadow_DOM.get_root_node_locs"
Setcompr_eq_image: fixes f :: "'c::type \<Rightarrow> 'b::type"   and A :: "'c::type set" shows "{(f::'c::type \<Rightarrow> 'b::type) x |x::'c::type. x \<in> (A::'c::type set)} = f ` A"
preserves_ide: fixes a :: "'c" assumes "C.ide a" shows "D.ide (G a)"
ternary_ternary_eval_idempotence_Not: fixes a :: "ternaryformula" shows "ternary_ternary_eval (TernaryNot (TernaryNot a)) = ternary_ternary_eval a"
divides_trans: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "Group.monoid G"   and "a divides\<^bsub>G\<^esub> b"   and "b divides\<^bsub>G\<^esub> c"   and "a \<in> carrier G" shows "a divides\<^bsub>G\<^esub> c"
iso_iff: shows "(gp \<in> Group.iso (reduced_homology_group (int n - 1) (nsphere (n - Suc 0))) (relative_homology_group (int n) (subtopology (nsphere n) {x. x n \<le> 0}) {x. x n = 0})) = (carrier (relative_homology_group (int n) (subtopology (nsphere n) {x. x n \<le> 0}) {x. x n = 0}) \<subseteq> gp ` carrier (reduced_homology_group (int n - 1) (nsphere (n - Suc 0))) \<and> (\<forall>x\<in>carrier (reduced_homology_group (int n - 1) (nsphere (n - Suc 0))). gp x = \<one>\<^bsub>relative_homology_group (int n) (subtopology (nsphere n) {x. x n \<le> 0}) {x. x n = 0}\<^esub> \<longrightarrow> x = \<one>\<^bsub>reduced_homology_group (int n - 1) (nsphere (n - Suc 0))\<^esub>))"
lub_in_lattice: fixes S :: "'a set" assumes "S \<subseteq> A" shows "lub S cl \<in> A"
union: fixes M :: "'a multiset"   and N :: "'a multiset" shows "F (M + N) = F M \<^bold>* F N"
rdomp_def: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> assn"   and a :: "'c" shows "rdomp R a \<equiv> \<exists>h c. h \<Turnstile> R a c"
poly_sgn_eventually_at_bot: fixes p :: "real poly" shows "\<forall>\<^sub>F x in at_bot. sgn (poly p x) = sgn_neg_inf p"
prems_2: shows "list_all (\<lambda>P. index_free P i) (P # Ps)"
not_\<omega>_lt_1: shows "\<not> \<omega> < 1"
card_fst_bf: fixes M :: "nat set"   and q :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and j :: "nat"   and i :: "nat" assumes "infinite M" shows "finite (fst (bf M q (j, i))) \<and> card (fst (bf M q (j, i))) = q j i"
Suc_k_less_ncols: shows "Suc k < ncols A"
Units_inv_inv: fixes x :: "'a" assumes "x \<in> Units \<G>" shows "inv (inv x) = x"
Sup_dual_def_var: fixes X :: "'a set" shows "\<partial> (\<Squnion> X) = \<Sqinter> (\<partial> ` X)"
image_def: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "\<P> f A = {y. \<exists>x\<in>A. y = f x}"
mat_empty_column_length: fixes nr :: "nat"   and nc :: "nat"   and M :: "'b list list" assumes "mat nr nc M"   and "M = []" shows "nc = 0"
pick_sum_impl_def: fixes s0 :: "nat"   and l :: "nat dlist list" shows "pick_sum_impl s0 l \<equiv> rfoldl (\<lambda>s x. RETURN (the (ls.sel x (\<lambda>_. True))) \<bind> (\<lambda>y. RETURN (s + y))) s0 l"
mem'_eq: shows "subst (to_partial mem |` Y) mem' = mem'"
extensional_subset: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'d set" assumes "f \<in> extensional A"   and "A \<subseteq> B" shows "f \<in> extensional B"
my_trancl_big_step_induct3: fixes ax :: "'a"   and ay :: "'b"   and az :: "'c"   and bx :: "'a"   and "by" :: "'b"   and bz :: "'c"   and r :: "((('a \<times> 'b) \<times> 'c) \<times> ('a \<times> 'b) \<times> 'c) set"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "(((ax, ay), az), (bx, by), bz) \<in> r\<^sup>+"   and "\<And>a b ba aa bb bc. (((a, b), ba), (aa, bb), bc) \<in> r \<Longrightarrow> P a b ba aa bb bc"   and "\<And>a b ba aa bb bc ab bd be. \<lbrakk>(((a, b), ba), (aa, bb), bc) \<in> r\<^sup>+; P a b ba aa bb bc; (((aa, bb), bc), (ab, bd), be) \<in> r; P aa bb bc ab bd be\<rbrakk> \<Longrightarrow> P a b ba ab bd be" shows "P ax ay az bx by bz"
span_carrier_lin_indpt_card_n: fixes W :: "'a vec set" assumes "W \<subseteq> carrier_vec n"   and "card W = n"   and "lin_indpt W" shows "local.span W = carrier_vec n"
RHS: shows "ac_Zp x k = (THE u. u \<in> carrier (residue_ring (p ^ k)) \<and> u * p ^ m = x (m + k))"
advantage_def: fixes \<A> :: "(bool, bool list, bool list) gpv" shows "PRF.advantage \<A> = \<bar>spmf (PRF.game_0 \<A>) True - spmf (PRF.game_1 \<A>) True\<bar>"
that: fixes D' :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry" assumes "A \<turnstile> \<langle>l, D\<rangle> \<leadsto>\<^bsub>v,n\<^esub> \<langle>l', D'\<rangle>"   and "Z = [D']\<^bsub>v,n\<^esub>" shows "thesis"
psi_def: fixes j :: "nat"   and x :: "nat" shows "\<psi> j x \<equiv> Some (e_nth (local.prefixes j (Suc x)) x)"
equal_cblinfun_def: fixes M :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and N :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "equal_class.equal M N = (mat_of_cblinfun M = mat_of_cblinfun N)"
strictly_generalizes_def: fixes C :: "'a literal multiset"   and D :: "'a literal multiset" shows "strictly_generalizes C D = (generalizes C D \<and> \<not> generalizes D C)"
card_Diff1_less_iff: fixes A :: "'b set"   and x :: "'b" shows "(card (A - {x}) < card A) = (finite A \<and> x \<in> A)"
simps: fixes n :: "nat"   and l :: "cell list"   and r :: "cell list" shows "inv_end4 n (l, r) = (\<exists>any. 0 < n \<and> l = Oc \<up> n @ [Bk] \<and> r = any # Oc \<up> n)"
case_transfer: fixes R2 :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and S :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and R1 :: "'h \<Rightarrow> 'i \<Rightarrow> bool" shows "((R2 ===> S) ===> rel_nondetT' R1 R2 ===> S) case_nondetT case_nondetT"
invalidate_sqn: fixes dests :: "nat \<Rightarrow> nat option"   and dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "\<forall>rsn. dests dip = Some rsn \<longrightarrow> sqn rt dip \<le> rsn" shows "sqn rt dip \<le> sqn (invalidate rt dests) dip"
f_Exec_Stream_Acc_LocalState__State_Idle_eq: fixes l :: "nat"   and k :: "nat"   and localState :: "'d \<Rightarrow> 'e"   and trans_fun :: "'f message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'f message_af list"   and output_fun :: "'d \<Rightarrow> 'g message_af"   and c :: "'d" assumes "0 < l"   and "l \<le> k"   and "Exec_Equal_State localState trans_fun"   and "\<And>n. n < length xs \<Longrightarrow> State_Idle localState output_fun trans_fun (f_Exec_Comp_Stream_Acc_LocalState l localState trans_fun xs c ! n)" shows "f_Exec_Comp_Stream_Acc_LocalState k localState trans_fun xs c = f_Exec_Comp_Stream_Acc_LocalState l localState trans_fun xs c"
splitFace_delete_oldF: fixes f12 :: "face"   and f21 :: "face"   and newGraph :: "graph"   and g :: "graph"   and ram1 :: "nat"   and ram2 :: "nat"   and oldF :: "face"   and newVertexList :: "nat list" assumes "(f12, f21, newGraph) = splitFace g ram1 ram2 oldF newVertexList"   and "oldF \<noteq> f12"   and "oldF \<noteq> f21"   and "distinct (faces g)" shows "oldF \<notin> \<F> newGraph"
Plus_into_Times: fixes a1 :: "'a"   and a2 :: "'a"   and A :: "'a set"   and b1 :: "'b"   and b2 :: "'b"   and B :: "'b set" assumes "a1 \<noteq> a2 \<and> {a1, a2} \<subseteq> A"   and "b1 \<noteq> b2 \<and> {b1, b2} \<subseteq> B" shows "\<exists>f. inj_on f (A <+> B) \<and> f ` (A <+> B) \<subseteq> A \<times> B"
enat_cosplit: fixes P :: "'c \<Rightarrow> bool"   and c :: "'c"   and d :: "enat \<Rightarrow> 'c"   and n :: "enat" shows "P (case n of 0 \<Rightarrow> c | eSuc x \<Rightarrow> d x) = ((n = 0 \<longrightarrow> P c) \<and> (\<forall>m. n = eSuc m \<longrightarrow> P (d m)))"
set_difference: fixes xs :: "'c list"   and ys :: "'c list" assumes "distinct xs"   and "distinct ys"   and "sorted xs"   and "sorted ys" shows "set (List_local.difference xs ys) = set xs - set ys"
eql1: shows "st.pmat_extension (H_k k) = st.ptensor_mat (H_k k) (1\<^sub>m st.d2)"
cos_conv_cosh: fixes z :: "complex" shows "cos z = cosh (\<i> * z)"
snd_item_rule_x: shows "snd (item_rule x) = \<alpha> @ \<beta>"
Fourier_offset_term: fixes f :: "real \<Rightarrow> real"   and t :: "real"   and n :: "nat" assumes "f absolutely_integrable_on {- pi..pi}"   and "\<And>x. f (x + 2 * pi) = f x" shows "Fourier_coefficient (\<lambda>x. f (x + t)) (2 * n + 2) * trigonometric_set (2 * n + 2) 0 = Fourier_coefficient f (2 * n + 1) * trigonometric_set (2 * n + 1) t + Fourier_coefficient f (2 * n + 2) * trigonometric_set (2 * n + 2) t"
simps_1: fixes K\<^sub>1 :: "clock"   and \<tau> :: "'b tag_const"   and K\<^sub>2 :: "clock" shows "\<lbrakk> K\<^sub>1 sporadic \<tau> on K\<^sub>2 \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = {\<rho>. \<exists>n. hamlet (Rep_run \<rho> n K\<^sub>1) \<and> time (Rep_run \<rho> n K\<^sub>2) = \<tau>}"
assms_3: shows "no_better_than y carrier relation \<subseteq> no_better_than x carrier relation"
comp_closed: fixes f :: "'d"   and g :: "'d" assumes "D.arr f \<and> Obj (src\<^sub>D f) \<and> Obj (trg\<^sub>D f)"   and "D.arr g \<and> Obj (src\<^sub>D g) \<and> Obj (trg\<^sub>D g)"   and "D.cod f = D.dom g" shows "D.seq g f \<and> Obj (src\<^sub>D (g \<cdot>\<^sub>D f)) \<and> Obj (trg\<^sub>D (g \<cdot>\<^sub>D f))"
finprod_const: fixes a :: "'a"   and A :: "'c set" assumes "a \<in> carrier R" shows "(\<Oplus>x\<in>A. a) = [card A] \<cdot> a"
to_fract_def: fixes x :: "'b" shows "to_fract x = Fraction_Field.Fract x (1::'b)"
simps_4: fixes f1 :: "'a"   and f2 :: "'a" shows "(case RECV of SEND \<Rightarrow> f1 | RECV \<Rightarrow> f2) = f2"
root_nemp: fixes u :: "'b list"   and r :: "'b list" assumes "u \<noteq> \<epsilon>"   and "u \<in> r*" shows "r \<noteq> \<epsilon>"
sums_cong: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b"   and c :: "'b" assumes "\<And>n. f n = g n" shows "f sums c = g sums c"
set_integrable_cong: fixes M :: "'a measure"   and M' :: "'a measure"   and A :: "'a set"   and A' :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b" assumes "M = M'"   and "A = A'"   and "\<And>x. x \<in> A \<Longrightarrow> f x = f' x" shows "set_integrable M A f = set_integrable M' A' f'"
node_total_order_trans: fixes e1 :: "'a event"   and i :: "nat"   and e2 :: "'a event"   and e3 :: "'a event" assumes "e1 \<sqsubset>\<^sup>i e2"   and "e2 \<sqsubset>\<^sup>i e3" shows "e1 \<sqsubset>\<^sup>i e3"
ereal_MInfty_eq_plus: fixes a :: "ereal"   and b :: "ereal" shows "(- \<infinity> = a + b) = (a = - \<infinity> \<and> b \<noteq> \<infinity> \<or> b = - \<infinity> \<and> a \<noteq> \<infinity>)"
interleave_filtered: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> bool" shows "xs \<in> filter P xs \<otimes> filter (\<lambda>x'. \<not> P x') xs"
snd_foldl_ef_det_eq: fixes bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and n :: "'b"   and A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat" shows "snd (foldl (echelon_form_of_column_k_det bezout) (n, A, 0) [0..<k]) = foldl (echelon_form_of_column_k bezout) (A, 0) [0..<k]"
prems: shows "a \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>"
fmerge_def: fixes return :: "'e fset \<Rightarrow> 'd"   and bind :: "'d \<Rightarrow> ('e fset \<Rightarrow> 'd) \<Rightarrow> 'd"   and A :: "'e fset"   and f :: "'e \<Rightarrow> 'd" shows "fmerge return bind A f = ffold (\<lambda>m1 m2. bind m1 (\<lambda>A. bind m2 (\<lambda>B. return (A |\<union>| B)))) (return {||}) (f |`| A)"
Gauss_Jordan_column_k_det_P_iarrays_def: fixes A' :: "'b \<times> nat \<times> 'b iarray iarray"   and k :: "nat" shows "Gauss_Jordan_column_k_det_P_iarrays A' k = (let det_P = fst A'; i = fst (snd A'); A = snd (snd A') in if vector_all_zero_from_index (i, column_iarray k A) \<or> i = nrows_iarray A then (det_P, i, A) else let gauss = Gauss_Jordan_in_ij_det_P_iarrays A i k in (fst gauss * det_P, i + 1, snd gauss))"
incomplete_imp_incomp_block: fixes bl :: "'a set" assumes "bl \<in># \<B>" shows "incomplete_block bl"
simps_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 cs :: "'a list"   and as :: "'a list"   and a :: "'a" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main" shows "CFG.upd_rev_cs kind cs (as @ [a]) = (case kind a of Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs \<Rightarrow> case cs of [] \<Rightarrow> CFG.upd_rev_cs kind cs as | c' # cs' \<Rightarrow> CFG.upd_rev_cs kind cs' as | Q\<hookleftarrow>\<^bsub>p\<^esub>f \<Rightarrow> CFG.upd_rev_cs kind (a # cs) as | _ \<Rightarrow> CFG.upd_rev_cs kind cs as)"
wfE_min: fixes R :: "('a \<times> 'a) set"   and x :: "'a"   and Q :: "'a set" assumes "wf R"   and "x \<in> Q" obtains z :: "'a" where "z \<in> Q" and "\<And>y. (y, z) \<in> R \<Longrightarrow> y \<notin> Q"
ennreal_indicator: fixes A :: "'a set"   and x :: "'a" shows "ennreal (indicat_real A x) = indicator A x"
length_finite_permutations_of_multiset: fixes xs :: "'a list"   and A :: "'a multiset" assumes "xs \<in> permutations_of_multiset A" shows "length xs = size A"
finite_nempty_preorder_has_min: fixes B :: "'b set"   and R :: "('b \<times> 'b) set" assumes "finite B"   and "B \<noteq> {}"   and "refl_on B R"   and "trans R"   and "total_on B R" shows "\<exists>x\<in>B. \<forall>y\<in>B. y \<succeq>[R] x"
s_cond_def_1: fixes t :: "real" assumes "t \<le> 0" shows "s t = s0"
lt_l: fixes a :: "MIR.num" shows "isrlfm (rsplit lt a)"
nth_Cons: fixes x :: "'b"   and xs :: "'b List.list"   and n :: "nat" shows "(x # xs) ! n = (case n of 0 \<Rightarrow> x | Suc k \<Rightarrow> xs ! k)"
simps: fixes i :: "nat" shows "s.h i = (if i \<le> 1 then 1 else if i = 2 then lead_coeff (local.F 2) ^ (degree (local.F 1) - degree (local.F (Suc 1))) else lead_coeff (local.F i) ^ (degree (local.F (i - 1)) - degree (local.F (Suc (i - 1)))) / s.h (i - 1) ^ (degree (local.F (i - 1)) - degree (local.F (Suc (i - 1))) - 1))"
ivl_disj_un_one_3: fixes l :: "'e"   and u :: "'e" assumes "l \<le> u" shows "{..l} \<union> {l<..u} = {..u}"
comm_neutral: fixes a :: "'e" shows "a + (0::'e) = a"
np_specI: fixes np :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" assumes "\<And>gs bs hs data. snd ` set (np gs bs hs data) \<subseteq> set hs \<times> (set gs \<union> set bs \<union> set hs) \<and> set hs \<times> (set gs \<union> set bs) \<subseteq> snd ` set (np gs bs hs data) \<and> (\<forall>a b. a \<in> set hs \<longrightarrow> b \<in> set hs \<longrightarrow> a \<noteq> b \<longrightarrow> (a, b) \<in>\<^sub>p snd ` set (np gs bs hs data)) \<and> (\<forall>p q. (True, p, q) \<in> set (np gs bs hs data) \<longrightarrow> q \<in> set gs \<union> set bs)" shows "np_spec np"
vsv: fixes a :: "V"   and b :: "V"   and A :: "V"   and c :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r \<restriction>\<^sup>r\<^sub>\<circ> A"   and "\<langle>a, c\<rangle> \<in>\<^sub>\<circ> r \<restriction>\<^sup>r\<^sub>\<circ> A" shows "b = c"
nonzero_memE_2: fixes R :: "('a, 'b) ring_scheme"   and a :: "'a" assumes "domain R"   and "a \<in> nonzero R" shows "a \<noteq> \<zero>\<^bsub>R\<^esub>"
one_product_group: fixes I :: "'e set"   and G :: "'e \<Rightarrow> ('f, 'g) monoid_scheme" shows "\<one>\<^bsub>product_group I G\<^esub> = (\<lambda>i\<in>I. \<one>\<^bsub>G i\<^esub>)"
R_OK: fixes v :: "'c"   and v' :: "'c"   and k :: "'b"   and k' :: "'b" assumes "invar v"   and "invar v'"   and "R' (k, v) (k', v')" shows "R (k, \<alpha> v) (k', \<alpha> v')"
B: shows "y \<in> range p \<or> y \<in> range q"
comp_arr_inv: fixes f :: "'a"   and g :: "'a" assumes "C.inverse_arrows f g" shows "f \<cdot>\<^sub>C g = C.dom g"
frontier_retract_of_punctured_universe: fixes S :: "'a set"   and a :: "'a" assumes "convex S"   and "bounded S"   and "a \<in> interior S" shows "frontier S retract_of - {a}"