hcju/isabelleps huggingface dataset:
assms: shows "\<turnstile>\<^sub>3 {P} c { Q}"
pos_neg_in_imp_true: fixes A :: "'b"   and C :: "'b literal multiset"   and I :: "'b set" assumes "Pos A \<in># C"   and "Neg A \<in># C" shows "I \<Turnstile> C"
simps_4: fixes y :: "'a" shows "flat_orD (deny (deny y)) = deny y"
triangle: fixes f :: "'b"   and g :: "'b" assumes "ide f"   and "ide g"   and "src g = trg f" shows "(g \<star> \<l>[f]) \<cdot> \<a>[g, src g, f] = \<r>[g] \<star> f"
sp_to_new_edge_tgt_imp: fixes g :: "('d \<times> nat, 'e) rgraph_scheme"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set"   and e :: "('d \<times> nat) edge"   and g' :: "('d \<times> nat, 'e) rgraph_scheme"   and v :: "'d \<times> nat"   and es :: "('d \<times> nat) edge list" assumes "sub_rel_of g subs"   and "ArcExt.extends g e g'"   and "SubRel.subpath g' v es (tgt e) subs"   and "v \<noteq> tgt e" shows "\<exists>es'. es = es' @ [e] \<and> e \<notin> set es' \<and> SubRel.subpath g v es' (src e) subs"
f_rec: fixes x :: "real" assumes "x\<^sub>1 < x" shows "f x = g x + (\<Sum>i<k. as ! i * f (bs ! i * x + (hs ! i) x))"
finite_mset_set_inter: fixes A :: "'c set"   and B :: "'c set" assumes "finite A"   and "finite B" shows "mset_set (A \<inter> B) = mset_set A \<inter># mset_set B"
commute: fixes a :: "'e"   and b :: "'e" shows "inf a b = inf b a"
\<K>_elem_reps_inj: fixes \<mu> :: "('s \<times> ('c \<Rightarrow> real) set) pmf"   and l :: "'s"   and R :: "('c \<Rightarrow> real) set" assumes "\<mu> \<in> \<K> (l, R)"   and "(l, R) \<in> \<S>" shows "inj_on reps (set_pmf \<mu>)"
fbrelation_vinsert: fixes a :: "V"   and b :: "V"   and r :: "V" shows "fbrelation (vinsert [a, b]\<^sub>\<circ> r) = fbrelation r"
fset_exhaust: fixes S :: "'b fset" obtains "S = {||}"   | x :: "'b" and S' :: "'b fset" where "S = finsert x S'"
bn_finite: fixes permute :: "perm \<Rightarrow> 'a \<Rightarrow> 'a"   and bn :: "'a \<Rightarrow> atom set"   and \<alpha> :: "'a" assumes "class.bn permute bn" shows "finite (bn \<alpha>)"
digit_decode_encode: fixes ds :: "nat list"   and a :: "nat" shows "digit_decode ds (digit_encode ds a) = a mod prod_list ds"
impI: fixes P :: "bool"   and Q :: "bool" assumes "P \<Longrightarrow> Q" shows "P \<longrightarrow> Q"
that: fixes x :: "'a" assumes "x \<in> X"   and "infdist y X = dist y x" shows "thesis"
sumSym: fixes Rel :: "(pi \<times> pi) set"   and P :: "pi"   and Q :: "pi" assumes "Id \<subseteq> Rel" shows "P \<oplus> Q \<leadsto>[Rel] Q \<oplus> P"
soundness_diagonalize_JNF: fixes diagonalize_JNF :: "'d mat \<Rightarrow> ('d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd) \<Rightarrow> 'd mat \<times> 'd mat \<times> 'd mat"   and m :: "nat"   and n :: "nat" assumes "Diagonalize.diagonalize diagonalize_JNF" shows "\<forall>A bezout. A \<in> carrier_mat m n \<and> is_bezout_ext bezout \<longrightarrow> (case diagonalize_JNF A bezout of (P, S, Q) \<Rightarrow> P \<in> carrier_mat m m \<and> Q \<in> carrier_mat n n \<and> S \<in> carrier_mat m n \<and> invertible_mat P \<and> invertible_mat Q \<and> isDiagonal_mat S \<and> S = P * A * Q)"
length_inf_drop_hint1: fixes s :: "nat \<Rightarrow> 'a list"   and k :: "nat" assumes "s k \<noteq> []" shows "length (inf_drop k s 0) \<noteq> 0"
sig_red_single_top_lt_rep_list: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and f :: "'t \<Rightarrow>\<^sub>0 'b"   and t :: "'a" assumes "sig_red_single sing_reg (=) p q f t"   and "rep_list q \<noteq> 0" shows "punit.lt (rep_list q) \<prec> punit.lt (rep_list p)"
insertDoubleAssertionStatEq: fixes F :: "'b frame"   and \<Psi> :: "'b"   and \<Psi>' :: "'b" shows "insertAssertion (insertAssertion F \<Psi>) \<Psi>' \<simeq>\<^sub>F insertAssertion F (\<Psi> \<otimes> \<Psi>')"
fresh_fmap_update: fixes a :: "atom"   and f :: "('c, 'd) fmap"   and x :: "'c"   and y :: "'d" assumes "a \<sharp> f"   and "a \<sharp> x"   and "a \<sharp> y" shows "a \<sharp> f(x $$:= y)"
iso_trans: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and I :: "('g, 'h) monoid_scheme" assumes "G \<cong> H"   and "H \<cong> I" shows "G \<cong> I"
R23_3: shows "pmf (sds R23) d = 1 - pmf (sds R23) a"
SUP_enat_add_left: fixes I :: "'c set"   and f :: "'c \<Rightarrow> enat"   and c :: "enat" assumes "I \<noteq> {}" shows "(\<Squnion>i\<in>I. f i + c) = \<Squnion> (f ` I) + c"
simps_5: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "subfrmlsn (\<phi> R\<^sub>n \<psi>) = {\<phi> R\<^sub>n \<psi>} \<union> subfrmlsn \<phi> \<union> subfrmlsn \<psi>"
trms\<^sub>s\<^sub>s\<^sub>t_mono: fixes M :: "('e, 'f) stateful_strand_step list"   and N :: "('e, 'f) stateful_strand_step list" assumes "set M \<subseteq> set N" shows "trms\<^sub>s\<^sub>s\<^sub>t M \<subseteq> trms\<^sub>s\<^sub>s\<^sub>t N"
minus_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<ominus> y \<in> carrier G"
Dom_cod: fixes f :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr" assumes "Cop_S.arr f" shows "Cop_S.Dom (Cop_S.cod f) = Cop_S.Cod f"
fminus_idemp: fixes A :: "'c fset"   and B :: "'c fset" shows "A |-| B |-| B = A |-| B"
gcsort_sort_intro: fixes key :: "'c \<Rightarrow> 'd"   and t :: "nat \<times> nat list \<times> 'c list"   and n :: "nat" assumes "sort_inv key t"   and "add_inv n t"   and "find ((<) (Suc 0)) (fst (snd t)) = None" shows "sorted (map key (gcsort_out t))"
resInput: fixes x :: "name"   and a :: "name"   and y :: "name"   and P :: "pi" assumes "x \<noteq> a"   and "x \<noteq> y" shows "<\<nu>x>a<y>.P \<sim> a<y>.(<\<nu>x>P)"
sphere_bounds': fixes x :: "real"   and y :: "real"   and z :: "real" assumes "x * x + y * y + z * z = 1" shows "- 1 \<le> x \<and> x \<le> 1"
rpsem_Test_simps: fixes \<phi> :: "formula"   and \<nu> :: "fin_string \<Rightarrow> real"   and b :: "fin_string \<Rightarrow> real" shows "([? \<phi>]\<nu> \<downharpoonleft> b) = (([\<phi>]\<nu> \<downharpoonright> True) \<and> \<nu> = b)"
P2_adv_bound: fixes D2 :: "bool \<times> 'index \<times> bool \<times> bool \<Rightarrow> bool spmf"   and b0 :: "bool"   and b1 :: "bool"   and \<sigma> :: "bool" assumes "\<forall>a. lossless_spmf (D2 a)" shows "\<bar>spmf (R2 (b0, b1) \<sigma> \<bind> D2) True - spmf (funct_OT_12 (b0, b1) \<sigma> \<bind> (\<lambda>(out1, out2). S2 \<sigma> out2 \<bind> D2)) True\<bar> \<le> \<bar>2 * (spmf (local.etp.HCP_game \<A> \<sigma> (if \<sigma> then b1 else b0) D2) True - 1 / 2)\<bar>"
univalent_def: fixes R :: "('e \<times> 'f) set" shows "univalent R = (\<forall>x y z. (x, y) \<in> R \<and> (x, z) \<in> R \<longrightarrow> z = y)"
rec: fixes n :: "nat" assumes "length fs - length cs \<le> n" shows "(\<Sum>k<length cs. cs ! k * f (n + k)) = (0::'a)"
lsc_liminf_at_eq: fixes x0 :: "'b"   and f :: "'b \<Rightarrow> ereal" shows "lsc_at x0 f = (f x0 = min (f x0) (Liminf (at x0) f))"
eq: shows "ideal F \<div> monomial (1::'a) t = UNIV"
cases: fixes a1 :: "'a list"   and a2 :: "'b sequent" assumes "upRulesp a1 a2" obtains c :: "'b sequent" and R :: "'b" and Fs :: "'b form list" and ps :: "'a list" where "a1 = ps" and "a2 = c" and "MultiSequents.mset c \<equiv> \<LM> Compound R Fs  \<RM>" and "ps \<noteq> []"
nat_induct_step2: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "P 0"   and "P 1"   and "\<And>m. P m \<Longrightarrow> P (Suc (Suc m))" shows "P n"
assms_2: shows "prec_spec h (R a p) (R a' p)"
v'_def: shows "v' \<equiv> b * f - t * #v"
Gramian_determinant_def: fixes G :: "'b vec list"   and k :: "nat" shows "gs.Gramian_determinant G k = det (gs.Gramian_matrix G k)"
reachable_del_vertI: fixes v :: "'a"   and w :: "'a" assumes "v \<rightarrow>\<^sup>*\<^bsub>G\<^esub> w"   and "u \<noteq> v \<or> u \<noteq> w" shows "v \<rightarrow>\<^sup>*\<^bsub>del_vert u\<^esub> w"
If2: fixes b :: "bexp"   and l\<^sub>1 :: "nat"   and l\<^sub>2 :: "nat"   and c\<^sub>1 :: "com"   and c\<^sub>2 :: "com" assumes "sec b \<le> min l\<^sub>1 l\<^sub>2"   and "\<turnstile> c\<^sub>1 : l\<^sub>1"   and "\<turnstile> c\<^sub>2 : l\<^sub>2" shows "\<turnstile> IF b THEN c\<^sub>1 ELSE c\<^sub>2 : min l\<^sub>1 l\<^sub>2"
hyps_4: fixes fs :: "(char list \<times> trm) list" assumes "[] \<turnstile> fs [:] fTs"   and "\<forall>a\<in>set fs. case a of (l, t) \<Rightarrow> t \<in> value" shows "\<exists>a. \<turnstile> fps [\<rhd>] fs \<Rightarrow> a"
finite_reduced_GB_Polys: fixes X :: "'x set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite X"   and "F \<subseteq> P[X]" shows "finite (i.punit.reduced_GB F)"
tau_def: fixes R :: "('g, 'h) Ring_scheme"   and M :: "('e, 'g, 'i) Module_scheme"   and N :: "('f, 'g, 'i) Module_scheme"   and P :: "('e \<times> 'f, 'g, 'i) Module_scheme" shows "tau R M N P = (\<lambda>x\<in>M \<times>\<^sub>c N. x)"
has_integral_combine_tagged_division: fixes p :: "('a \<times> 'a set) set"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and i :: "'a set \<Rightarrow> 'b" assumes "p tagged_division_of S"   and "\<And>x k. (x, k) \<in> p \<Longrightarrow> (f has_integral i k) k" shows "(f has_integral (\<Sum>(x, k)\<in>p. i k)) S"
sum_inverse_squares_expansion_bigo: fixes N :: "nat" shows "(\<lambda>n. (\<Sum>i = 1..n. 1 / (real i)\<^sup>2) - (pi\<^sup>2 / 6 - 1 / real n + 1 / (2 * (real n)\<^sup>2) - (\<Sum>i = 1..N. bernoulli (2 * i) / real n ^ (2 * i + 1)))) \<in> O(\<lambda>n. 1 / real n ^ (2 * N + 3))"
mem_ith: fixes ei :: "'a"   and es :: "'a list" assumes "ei \<in> set es" shows "\<exists>el er. es = el @ ei # er"
card_Union_disjoint: fixes C :: "'c set set" assumes "pairwise disjnt C"   and "\<And>A. A \<in> C \<Longrightarrow> finite A" shows "card (\<Union> C) = sum card C"
augment_forest: fixes F :: "'edge set"   and e :: "'edge"   and u :: "'a"   and v :: "'a" assumes "forest F"   and "e \<in> E - F"   and "joins u v e" shows "forest (insert e F) = ((u, v) \<notin> connected F)"
dens_ctxt_measure_bind_const: fixes \<rho> :: "nat \<Rightarrow> val"   and N :: "'a measure" assumes "\<rho> \<in> space (state_measure V' \<Gamma>)"   and "subprob_space N" shows "dens_ctxt_measure \<Y> \<rho> \<bind> (\<lambda>_. N) = density N (\<lambda>_. branch_prob \<Y> \<rho>)"
span_image: fixes s1 :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and s2 :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> 'e"   and S :: "'d set" assumes "module_hom s1 s2 f" shows "module.span s2 (f ` S) = f ` module.span s1 S"
bij_betw_chambers: fixes C :: "'a set" assumes "chamber C" shows "bij_betw f C (f ` C)"
Nil_is_append_conv: fixes xs :: "'a list"   and ys :: "'a list" shows "([] = append xs ys) = (xs = [] \<and> ys = [])"
assms_1: shows "folding_g.label_wrt B \<phi>"
A: fixes a :: "'a" assumes "Pre sc1 (S a)" shows "Pre sc2 a"
sum_list_replicate: fixes n :: "nat"   and c :: "nat" shows "sum_list (replicate n c) = n * c"
abelian_monoidE_1: fixes R :: "('h, 'i) ring_scheme"   and x :: "'h"   and y :: "'h" assumes "abelian_monoid R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y \<in> carrier R"
ln_eq_zero_iff: fixes x :: "real" assumes "0 < x" shows "(ln x = 0) = (x = 1)"
is_density_expr_def: shows "is_density_expr \<equiv> \<lambda>(vs, vs', \<Gamma>, \<delta>) t e. case_nat t \<Gamma> \<turnstile>\<^sub>c e : REAL \<and> free_vars e \<subseteq> shift_var_set (set vs') \<and> nonneg_cexpr (shift_var_set (set vs')) (case_nat t \<Gamma>) e"
vsubset_vdoubleton_leftD_2: fixes a :: "V"   and b :: "V"   and A :: "V" assumes "set {a, b} \<subseteq>\<^sub>\<circ> A" shows "b \<in>\<^sub>\<circ> A"
set_interact: fixes xs :: "'a list list"   and ys :: "'a list list" shows "list.set (interact xs ys) = list.set (concat xs) \<union> list.set (concat ys)"
rTau: fixes Q' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> Q \<longmapsto> \<tau> \<prec> Q'" shows "\<exists>P'. \<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P' \<and> (\<Psi>, P', Q') \<in> Rel"
less_add_same_cancel2: fixes a :: "'b multiset"   and b :: "'b multiset" shows "(a \<subset># b + a) = ({#} \<subset># b)"
concurrent_operations_commute: fixes xs :: "('a \<times> ('a, 'b) operation) event list"   and i :: "nat" assumes "xs prefix of i" shows "hb.concurrent_ops_commute (node_deliver_messages xs)"
finter_fminus: fixes A :: "'b fset"   and B :: "'b fset"   and C :: "'b fset" shows "A |\<inter>| B |-| C = A |\<inter>| (B |-| C)"
valid_decomp_Nil: fixes X :: "'x set" shows "valid_decomp X []"
mult_sv_ex_id2: fixes v :: "complex \<times> complex" shows "\<exists>k. k \<noteq> 0 \<and> v = k *\<^sub>s\<^sub>v v"
I: fixes j :: "nat" assumes "j \<in> {n - 2 * N..<a * N}" shows "u N ((T ^^ j) x) \<le> E2"
total_onI: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> (x, y) \<in> r \<or> x = y \<or> (y, x) \<in> r" shows "total_on A r"
is_poincare_line_mk_circline: fixes A :: "complex"   and B :: "complex"   and C :: "complex"   and D :: "complex" assumes "(A, B, C, D) \<in> hermitean_nonzero" shows "is_poincare_line (mk_circline A B C D) = ((cmod A)\<^sup>2 < (cmod B)\<^sup>2 \<and> A = D)"
incidence_system_axioms: shows "incidence_system \<U> \<A>"
intros_7: fixes X :: "freemsg"   and Y :: "freemsg" assumes "X \<sim> Y" shows "Y \<sim> X"
injectivity_2: fixes X :: "'a set"   and Y :: "'a set"   and k :: "nat"   and n :: "nat"   and k' :: "nat"   and xs :: "'a list"   and xs' :: "'a list"   and ys :: "'a list"   and ys' :: "'a list"   and zs :: "'a list"   and zs' :: "'a list" assumes "X \<inter> Y = {}"   and "k \<in> {0..n} \<and> k' \<in> {0..n}"   and "(length xs = k \<and> distinct xs \<and> set xs \<subseteq> X) \<and> length xs' = k' \<and> distinct xs' \<and> set xs' \<subseteq> X"   and "(length ys = n - k \<and> distinct ys \<and> set ys \<subseteq> Y) \<and> length ys' = n - k' \<and> distinct ys' \<and> set ys' \<subseteq> Y"   and "interleavings xs ys zs \<and> interleavings xs' ys' zs'"   and "zs = zs'" shows "xs = xs'"
intro: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool" assumes "FullModel wtFsym wtPsym arOf resOf parOf \<Phi> intT intF intP" shows "M_FullModel wtFsym wtPsym arOf resOf parOf \<Phi> intT intF intP"
P1: shows "Coplanar A B C T \<and> Bet P T Q"
cancel_div_mod_rules_2: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and 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 modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and b :: "'c"   and a :: "'c"   and c :: "'c" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "plus (plus (times b (divide a b)) (modulo a b)) c = plus a c"
lmerge_def: fixes A :: "'a list"   and f :: "'a \<Rightarrow> 'm" shows "local.lmerge A f = lUnionM (map f A)"
ket_injective: fixes i :: "'c"   and j :: "'c" shows "(ket i = ket j) = (i = j)"
seqstar_inductl: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z + x ; y \<le> y" shows "x\<^sup>; ; z \<le> y"
right_total_alt_def: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "right_total R = ((=) \<le> R\<inverse>\<inverse> OO R)"
lfilter_LCons: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b llist" shows "lfilter P (LCons x xs) = (if P x then LCons x (lfilter P xs) else lfilter P xs)"
individually_rational_on_def: fixes ds :: "D2 set"   and X :: "X5 set" shows "BSI.individually_rational_on ds X = (BSI.CD_on ds X = X \<and> BSI.CH X = X)"
iNextStrong_imp_iNextWeak: fixes t0 :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "iNextStrong t0 I P \<longrightarrow> iNextWeak t0 I P"
\<psi>_bigo: shows "primes_psi \<in> O(\<lambda>x. x)"
hd_append: fixes xs :: "'e list"   and ys :: "'e list" shows "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
intro: fixes s_ops :: "('a, 'b, 'c) set_ops_scheme"   and l_ops :: "('a, 'd, 'e) list_ops_scheme" assumes "StdSet s_ops"   and "StdList l_ops" shows "set_to_list_loc s_ops l_ops"
simps_1: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and uu :: "'d list" shows "filter2 P [] uu = ([], [])"
scale_one: fixes x :: "'b" shows "(1::'a) *a x = x"
bij_comp: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'f" assumes "bij f"   and "bij g" shows "bij (g \<circ> f)"
P_def: shows "P = greedy_conforming_path p \<sigma> \<sigma>' v0"
sup_top_left: fixes inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and top :: "'b::type"   and x :: "'b::type" assumes "class.bounded_lattice_top (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (top::'b::type)" shows "(sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (top::'b::type) (x::'b::type) = top"
pmat_extension_def: fixes m :: "'a mat" shows "stm2.pmat_extension m = stm2.ptensor_mat m (1\<^sub>m stm2.d2)"
conj_cancel_left: fixes x :: "'a" shows "\<sim> x \<^bold>\<squnion> x = \<one>"
functor_axioms: shows "functor (\<cdot>\<^sub>B) (\<cdot>\<^sub>C) G'"
MInfty_neq_ereal_1: fixes r :: "real" shows "ereal r \<noteq> - \<infinity>"
B: shows "\<forall>a. (a < n) = (a < m)"
sccs_verts_def: shows "sccs_verts = {S. S \<noteq> {} \<and> (\<forall>u\<in>S. \<forall>v\<in>S. u \<rightarrow>\<^sup>* v) \<and> (\<forall>u\<in>S. \<forall>v. v \<notin> S \<longrightarrow> \<not> u \<rightarrow>\<^sup>* v \<or> \<not> v \<rightarrow>\<^sup>* u)}"
P1: shows "Bet B A A0 \<and> Cong A A0 B' A' \<and> Bet B C C0 \<and> Cong C C0 B' C' \<and> Bet B' A' D0 \<and> Cong A' D0 B A \<and> Bet B' C' F0 \<and> Cong C' F0 B C \<and> Cong A0 C0 D0 F0"
are_open_subspaces: fixes X :: "'a set"   and is_open :: "'a set \<Rightarrow> bool"   and U :: "'a set"   and I :: "real set"   and C :: "real \<Rightarrow> 'a set"   and i :: "real" assumes "open_cover_of_subset X is_open U I C"   and "i \<in> I" shows "is_open (C i)"
fls_const_def: fixes c :: "'a" shows "fls_const c \<equiv> Abs_fls (\<lambda>n. if n = 0 then c else (0::'a))"
betw_c_in_path: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and ab :: "'b set" assumes "MinkowskiBetweenness \<E> \<P> betw"   and "betw a b c"   and "ab \<in> \<P> \<and> a \<in> ab \<and> b \<in> ab \<and> a \<noteq> b" shows "c \<in> ab"
null_char: shows "A_BxA.null = (A_B.null, A.null)"
Q''Chain: shows "Q'' \<Longrightarrow>\<^sub>\<tau> Q'"
finite_bind: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c set" assumes "finite S"   and "\<forall>x\<in>S. finite (f x)" shows "finite (Set.bind S f)"
canonic_int_ordered_def: fixes atom_ordering :: "('a \<times> 'a) set"   and S :: "'a Literal set set" assumes "propositional_atoms atom_ordering" shows "propositional_atoms.canonic_int_ordered atom_ordering S = {A. propositional_atoms.canonic_int_fun_ordered atom_ordering A S}"
homotopic_with_canon_on_empty: fixes t :: "'b set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "homotopic_with_canon (\<lambda>x. True) {} t f g"
paths_eq_def: fixes k :: "nat"   and P :: "(symbol \<times> character list) list set" shows "paths_eq k P = {p \<in> P. charslength p = k}"
bin_to_bl_minus1_aux: fixes n :: "nat"   and bl :: "bool list" shows "bin_to_bl_aux n (- 1) bl = replicate n True @ bl"
andfold_MatchExp_not_disc_negatedI: fixes ms :: "'c match_expr list"   and disc :: "'c \<Rightarrow> bool"   and neg :: "bool" assumes "\<forall>m\<in>set ms. \<not> has_disc_negated disc neg m" shows "\<not> has_disc_negated disc neg (andfold_MatchExp ms)"
PSize: shows "P \<in> subgroups_of_size (p ^ a)"
le_cong_r: fixes L :: "('a, 'b) gorder_scheme"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "weak_partial_order L"   and "x \<sqsubseteq>\<^bsub>L\<^esub> y"   and "y .=\<^bsub>L\<^esub> z"   and "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<sqsubseteq>\<^bsub>L\<^esub> z"
comp_def: fixes y :: "'a"   and x :: "'a" shows "y \<cdot> x \<equiv> if x \<in> Obj \<and> x = y then x else Null"
additive_content_division: fixes d :: "'c set set"   and a :: "'c"   and b :: "'c" assumes "d division_of cbox a b" shows "sum content d = content (cbox a b)"
eq_from_eq_frac: fixes r :: "'a"   and s :: "'a"   and r' :: "'a"   and s' :: "'a" assumes "local.frac r s = local.frac r' s'"   and "s \<in> R\<setminus>\<pp>"   and "s' \<in> R\<setminus>\<pp>"   and "r \<in> R"   and "r' \<in> R" obtains h :: "'a" where "h \<in> R\<setminus>\<pp>" and "h \<cdot> (s' \<cdot> r - s \<cdot> r') = \<zero>"
unclocked_single_list_sound: fixes v :: "v sem_env"   and s :: "'b state"   and es :: "exp0 list"   and bv :: "'b state \<times> (v list, v) result" assumes "Big_Step_Unclocked_Single.evaluate_list (Big_Step_Unclocked.evaluate v) s es bv" shows "Big_Step_Unclocked.evaluate_list v s es bv"
mset_factorsE: fixes F :: "'a multiset"   and p :: "'a" assumes "mset_factors F p" obtains "\<And>f. f \<in># F \<Longrightarrow> irreducible f" and "F \<noteq> {#}" and "\<Prod>\<^sub># F = p"
sum_list_Leq_Rel: fixes f :: "'a \<Rightarrow> 'b le_constraint"   and C :: "'a list" shows "((\<Sum>x\<leftarrow>C. lec_rel (f x)) = Leq_Rel) = (\<forall>x\<in>set C. lec_rel (f x) = Leq_Rel)"
le: shows "\<forall>i\<in>{i. n cd\<^bsup>\<pi>\<^esup>\<rightarrow> i \<and> i < k}. \<forall>j\<in>{i. n cd\<^bsup>\<pi>\<^esup>\<rightarrow> i \<and> k \<le> i}. i < j"
of_int_rat: fixes k :: "int" shows "rat_of_int k = Rat.Fract k 1"
finite_ordering_inj_on: fixes X :: "'a set"   and f :: "nat \<Rightarrow> 'a"   and ord :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "finite X"   and "TernaryOrdering.ordering f ord X" shows "inj_on f {0..card X - 1}"
countable_SIGMA: fixes I :: "'c set"   and A :: "'c \<Rightarrow> 'd set" assumes "countable I"   and "\<And>i. i \<in> I \<Longrightarrow> countable (A i)" shows "countable (Sigma I A)"
inverse_arrows_def: fixes f :: "'a"   and g :: "'a" shows "inverse_arrows f g \<equiv> ide (g \<cdot> f) \<and> ide (f \<cdot> g)"
zipWithS_fix_ind_lemma: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and i :: "nat"   and j :: "nat" assumes "\<And>j. P 0 j"   and "\<And>i j. \<lbrakk>P i j; Q i j\<rbrakk> \<Longrightarrow> P (Suc i) j"   and "\<And>i. Q i 0"   and "\<And>i j. \<lbrakk>P i j; Q i j\<rbrakk> \<Longrightarrow> Q i (Suc j)" shows "P i j \<and> Q i j"
arev_eq_iff: fixes G :: "('e, 'f) pre_digraph"   and arev :: "'f \<Rightarrow> 'f"   and x :: "'f"   and y :: "'f" assumes "bidirected_digraph G arev" shows "(arev x = arev y) = (x = y)"
one_matrix_def: shows "mone = (\<lambda>(i, j). if i = j then 1::'d else bot)"
gen_new_keys_not_analzd: fixes K :: "nat"   and evs :: "event list"   and KK :: "nat set" assumes "Key K \<notin> used evs"   and "K \<in> symKeys"   and "evs \<in> set_cr" shows "K \<notin> keysFor (analz (Key ` KK \<union> knows Spy evs))"
cases: fixes a1 :: "'b sequent list \<times> 'b sequent"   and a2 :: "'b form" assumes "leftPrincipal a1 a2" obtains C :: "'b sequent" and F :: "'b" and Fs :: "'b form list" and Ps :: "'b sequent list" where "a1 = (Ps, C)" and "a2 = Compound F Fs" and "C = ( \<LM> Compound F Fs  \<RM> \<Rightarrow>* \<Empt>)"
filter_Cons2: fixes x :: "'a"   and ys :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "x \<notin> set ys" shows "filter (\<lambda>y. y = x \<or> P y) ys = filter P ys"
right_assoc: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ac_operator f" shows "f (f a b) c = f a (f b c)"