hcju/isabelleps huggingface dataset:
select_convs_1: fixes ip :: "nat"   and sn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rreqs :: "(nat \<times> nat) set"   and store :: "nat \<Rightarrow> (p \<times> nat list) option"   and msg :: "msg"   and data :: "nat"   and dests :: "nat \<Rightarrow> nat option"   and pre :: "nat set"   and rreqid :: "nat"   and dip :: "nat"   and oip :: "nat"   and hops :: "nat"   and dsn :: "nat"   and dsk :: "k"   and osn :: "nat"   and sip :: "nat"   and handled :: "bool"   and more :: "'a" shows "ip \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, pre = pre, rreqid = rreqid, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, handled = handled, \<dots> = more\<rparr> = ip"
feq: fixes u :: "complex"   and z :: "complex" assumes "cmod u = 1"   and "z \<in> s" shows "f (u * z) = u * f z"
cinsertI1: fixes a :: "'f"   and B :: "'f cset" shows "cin a (cinsert a B)"
Theorem_6_fp_cop_F_1: fixes ds :: "'b set"   and X :: "'a set"   and d :: "'b" assumes "Singleton_for_d.stable_on ds X"   and "d \<in> ds" shows "card (Singleton_for_d.Cd d X) = card (Singleton_for_d.Cd d (Singleton_for_d.cop ds))"
abs_eq: fixes x :: "'k \<Rightarrow> nat"   and xa :: "'k" assumes "eq_onp (\<lambda>x. x \<in> multiset) x x" shows "add_mset xa (Abs_multiset x) = Abs_multiset (\<lambda>b. if b = xa then Suc (x b) else x b)"
trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p_funs_term_cases: fixes t :: "('c, 'd) Term.term"   and s :: "('c, 'd) stateful_strand_step"   and \<theta> :: "'d \<Rightarrow> ('c, 'd) Term.term"   and f :: "'c" assumes "t \<in> trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p (s \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>)"   and "f \<in> funs_term t" shows "(\<exists>u\<in>trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p s. f \<in> funs_term u) \<or> (\<exists>x\<in>fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p s. f \<in> funs_term (\<theta> x))"
card_UNIV_uprod: shows "CARD('a uprod) = CARD('a) * (CARD('a) + 1) div 2"
zero_diff: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and a :: "'e" assumes "class.comm_monoid_diff plus minus zero" shows "minus zero a = zero"
assms_1: shows "\<forall>j\<le>D. str (t + j + (i + k)) = x"
matrix_vector_mul_component: fixes A :: "((real, 'b) vec, 'a) vec"   and x :: "(real, 'b) vec"   and k :: "'a" shows "(A *v x) $ k = A $ k \<bullet> x"
opt_mru_init_def: shows "opt_mru_init = {\<lparr>opt_mru_state.next_round = 0, mru_vote = Map.empty, decisions = Map.empty\<rparr>}"
nsplitTree_correct: fixes n :: "nat"   and s :: "('b, 'c) FingerTreeStruc"   and p :: "'c \<Rightarrow> bool"   and i :: "'c"   and l :: "('b, 'c) FingerTreeStruc"   and nd :: "('b, 'c) Node"   and r :: "('b, 'c) FingerTreeStruc" assumes "is_leveln_ftree n s"   and "is_measured_ftree s"   and "\<And>a b. p a \<Longrightarrow> p (a + b)"   and "\<not> p i"   and "p (i + gmft s)"   and "nsplitTree p i s = (l, nd, r)" shows "toList s = toList l @ nodeToList nd @ toList r \<and> \<not> p (i + gmft l) \<and> p (i + gmft l + gmn nd) \<and> is_leveln_ftree n l \<and> is_measured_ftree l \<and> is_leveln_ftree n r \<and> is_measured_ftree r \<and> is_leveln_node n nd \<and> is_measured_node nd"
lm014: fixes a :: "('a \<times> 'b set) set"   and G :: "'b set"   and N :: "'a set" assumes "a \<in> allAllocations N G"   and "finite G" shows "\<forall>y\<in>Range a. finite y"
le: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" shows "\<bar>dist a b - dist c d\<bar> \<le> dist a c + dist b d"
proj_psi_lR_mult_post: shows "proj_psi_l R * post = proj_psi_l R"
finite_ReZ_segments_linepath: fixes a :: "complex"   and b :: "complex"   and z :: "complex" shows "finite_ReZ_segments (linepath a b) z"
type_universe_eq_imp_type_eq: fixes t1 :: "pdf_type"   and t2 :: "pdf_type" assumes "type_universe t1 = type_universe t2" shows "t1 = t2"
rAlpha: fixes \<alpha> :: "'a action"   and C :: "'d"   and p :: "(name \<times> name) list"   and P' :: "('a, 'b, 'c) psi" assumes "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "bn \<alpha> \<sharp>* C"   and "set p \<subseteq> set (bn \<alpha>) \<times> set (bn (p \<bullet> \<alpha>))"   and "distinctPerm p"   and "bn (p \<bullet> \<alpha>) \<sharp>* \<alpha>"   and "bn (p \<bullet> \<alpha>) \<sharp>* P'"   and "Prop C \<Psi> (P \<parallel> !P) \<alpha> P'" shows "Prop C \<Psi> (P \<parallel> !P) (p \<bullet> \<alpha>) (p \<bullet> P')"
extreme_point_of_empty: fixes x :: "'b" shows "\<not> x extreme_point_of {}"
ode_sem_eq: fixes ODE :: "('d, 'e) ODE"   and \<phi> :: "('d, 'f, 'e) formula"   and I :: "('d, 'f, 'e) interp" assumes "osafe ODE"   and "fsafe \<phi>" shows "{(\<nu>, mk_v I ODE \<nu> (sol t)) |\<nu> sol t. 0 \<le> t \<and> (sol solves_ode (\<lambda>_. ODE_sem I ODE)) {0..t} {x. mk_v I ODE \<nu> x \<in> fml_sem I \<phi>} \<and> VSagree (sol 0) (fst \<nu>) {x |x. Inl x \<in> FVP (EvolveODE ODE \<phi>)}} = {(\<nu>, mk_v I ODE \<nu> (sol t)) |\<nu> sol t. 0 \<le> t \<and> (sol solves_ode (\<lambda>_. ODE_sem I ODE)) {0..t} {x. mk_v I ODE \<nu> x \<in> fml_sem I \<phi>} \<and> sol 0 = fst \<nu>}"
p12_def: shows "p12 = coeff (pCons (rat_of_int x) b2) (degree (of_int_poly q)) / lead_coeff (of_int_poly q)"
UN_Un_distrib: fixes A :: "'c \<Rightarrow> 'b set"   and B :: "'c \<Rightarrow> 'b set"   and I :: "'c set" shows "(\<Union>i\<in>I. A i \<union> B i) = \<Union> (A ` I) \<union> \<Union> (B ` I)"
list_of_eucl_eucl_of_list: fixes xs :: "real list" assumes "length xs = DIM('e)" shows "list_of_eucl (eucl_of_list xs) = xs"
bot_fun_def: shows "Orderings.bot_class.bot = (\<lambda>x. Orderings.bot_class.bot)"
inf_prefix_length_maximality2: fixes xs :: "'b list"   and ys :: "'b list"   and i :: "nat" assumes "inf_prefix_length xs ys \<noteq> min (length xs) (length ys)"   and "inf_prefix_length xs ys \<le> i" shows "\<exists>j\<le>i. xs ! j \<noteq> ys ! j"
measurable_sets_borel: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'d measure"   and A :: "'d set" assumes "f \<in> borel \<rightarrow>\<^sub>M M"   and "A \<in> sets M" shows "f -` A \<in> sets borel"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b list \<Rightarrow> 'a"   and f3 :: "'b list \<Rightarrow> 'b list \<Rightarrow> 'a"   and blue_witness :: "'b blue_witness" shows "P (case blue_witness of NO_CYC \<Rightarrow> f1 | REACH x xa \<Rightarrow> f2 x xa | CIRC x xa \<Rightarrow> f3 x xa) = (\<not> (blue_witness = NO_CYC \<and> \<not> P f1 \<or> (\<exists>x21 x22. blue_witness = REACH x21 x22 \<and> \<not> P (f2 x21 x22)) \<or> (\<exists>x31 x32. blue_witness = CIRC x31 x32 \<and> \<not> P (f3 x31 x32))))"
affine_dependent_biggerset: fixes s :: "'a set" assumes "finite s"   and "DIM('a) + 2 \<le> card s" shows "affine_dependent s"
disj_one_left: fixes x :: "'c" shows "- (1::'c) OR x = - (1::'c)"
singular_relboundary_minus: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" shows "singular_relboundary p X S (- c) = singular_relboundary p X S c"
downsetI: fixes xss :: "'a list set" assumes "\<And>xs. \<lbrakk>xs \<in> xss; xs \<noteq> []\<rbrakk> \<Longrightarrow> butlast xs \<in> xss" shows "downset xss"
if_smult: fixes P :: "bool"   and x :: "real"   and y :: "real"   and v :: "'a" shows "(if P then x else y) *\<^sub>R v = (if P then x *\<^sub>R v else y *\<^sub>R v)"
th3_6_1: fixes a :: "(real, 'a) vec"   and b :: "(real, 'a) vec"   and c :: "(real, 'a) vec"   and d :: "(real, 'a) vec" assumes "B\<^sub>\<real> a b c"   and "B\<^sub>\<real> a c d" shows "B\<^sub>\<real> b c d"
card_of_Un1: fixes A :: "'b set"   and B :: "'b set" shows "|A| \<le>o |A \<union> B|"
2: shows "\<exists>x\<in>X. n (S1 x \<triangle> S2 x) < n (S1 x)"
null_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e_def: shows "null \<equiv> Abs_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e \<lfloor>None\<rfloor>"
INTER_eq: fixes B :: "'c \<Rightarrow> 'b set"   and A :: "'c set" shows "\<Inter> (B ` A) = {y. \<forall>x\<in>A. y \<in> B x}"
lF_simps_1: fixes V\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>C :: "'d \<Rightarrow> 'd"   and src\<^sub>C :: "'d \<Rightarrow> 'd"   and trg\<^sub>C :: "'d \<Rightarrow> 'd"   and V\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and H\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<a>\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<i>\<^sub>D :: "'e \<Rightarrow> 'e"   and src\<^sub>D :: "'e \<Rightarrow> 'e"   and trg\<^sub>D :: "'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and \<Phi> :: "'d \<times> 'd \<Rightarrow> 'e"   and f :: "'d" assumes "pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>"   and "partial_magma.ide V\<^sub>C f" shows "partial_magma.arr V\<^sub>D (pseudofunctor.lF \<i>\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi> f)"
floor_add: fixes x :: "'a"   and y :: "'a" shows "\<lfloor>x + y\<rfloor> = (if frac x + frac y < (1::'a) then \<lfloor>x\<rfloor> + \<lfloor>y\<rfloor> else \<lfloor>x\<rfloor> + \<lfloor>y\<rfloor> + 1)"
distinct_1: fixes x1 :: "agent"   and x2 :: "nat" shows "msg.Agent x1 \<noteq> Number x2"
prime_aux: fixes p :: "nat"   and x :: "nat" assumes "prime p" shows "(if x = 0 then Suc 1 else (p ^ (Suc 1 * x) - 1) div (p ^ x - 1)) = p ^ x + 1"
get_trap_set_def: fixes state :: "'b sparc_state" shows "get_trap_set state \<equiv> traps state"
terminal_tinfinite: fixes xs :: "('c, 'd) tllist" assumes "\<not> tfinite xs" shows "terminal xs = undefined"
rel_converter_eq14: fixes B :: "'b \<Rightarrow> 'e \<Rightarrow> bool"   and C :: "'c \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_converter (=) B C (=) = rel_converter' B C"
MOST_I: fixes P :: "'b \<Rightarrow> bool" assumes "\<And>x. P x" shows "\<forall>\<^sub>\<infinity>x. P x"
zero_fun_def: shows "0 = (\<lambda>x. 0::'d)"
LeftDerivationIntro_cut_appendix': fixes \<alpha> :: "symbol list"   and \<delta> :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and ix :: "nat"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and j :: "nat"   and \<beta> :: "symbol list"   and \<delta>' :: "symbol list" assumes "LeftDerivationIntro (\<alpha> @ \<delta>) i r ix D j (\<beta> @ \<delta>')"   and "LeftDerivation \<alpha> ((i, r) # D) \<beta>"   and "i < length \<alpha>"   and "j < length \<beta>" shows "LeftDerivationIntro \<alpha> i r ix D j \<beta>"
intros_7: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and m :: "'Var ModeUpd" assumes "no_await c" shows "no_await (c@[m])"
rep_eq: fixes A :: "'b set" assumes "rp_rf A" shows "Rep_\<I> (rp_rf.\<I> A) = (\<lambda>x. if x \<in> A then A else {})"
simps: fixes a :: "fm" shows "(a \<in> boolean_axioms) = ((\<exists>A. a = A IMP A) \<or> (\<exists>A B. a = A IMP A OR B) \<or> (\<exists>A. a = A OR A IMP A) \<or> (\<exists>A B C. a = A OR B OR C IMP (A OR B) OR C) \<or> (\<exists>C A B. a = C OR A IMP (C IMP B) IMP A OR B))"
homeomorphic_map_def: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" shows "homeomorphic_map X Y f \<equiv> quotient_map X Y f \<and> inj_on f (topspace X)"
g: shows "finite {i \<in> I. g i \<noteq> 0}"
simps_2: fixes m :: "'a SecurityInvariant_configured"   and Ms :: "'a SecurityInvariant_configured list"   and Es :: "('a \<times> 'a) list"   and G :: "'a graph" shows "TopoS_Composition_Theory.generate_valid_topology_some (m # Ms) Es G = (if c_sinvar m G then TopoS_Composition_Theory.generate_valid_topology_some Ms Es G else FiniteGraph.delete_edges (TopoS_Composition_Theory.generate_valid_topology_some Ms Es G) (set (TopoS_withOffendingFlows.minimalize_offending_overapprox (c_sinvar m) Es [] G)))"
vD_def: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" shows "vD rt \<equiv> {dip. flag rt dip = Some val}"
single_uminus: fixes k :: "'c"   and a :: "'d" shows "monomial (- a) k = - monomial a k"
at_ide_is_diagram: fixes a :: "'c" assumes "Cop.ide a" shows "diagram J (\<cdot>\<^sub>S) (YoD.at a (local.map \<circ> D))"
sorted_Cons_le: fixes x :: "'b"   and xs :: "'b list"   and y :: "'b" assumes "ASSUMPTION (sorted (x # xs))"   and "y \<le> x" shows "sorted (y # xs)"
wiring_comp_converterI: fixes \<I> :: "('i, 'j) \<I>"   and \<I>' :: "('k, 'l) \<I>"   and conv1 :: "('i, 'j, 'k, 'l) converter"   and fg :: "('i \<Rightarrow> 'k) \<times> ('l \<Rightarrow> 'j)"   and \<I>'' :: "('m, 'n) \<I>"   and conv2 :: "('k, 'l, 'm, 'n) converter"   and fg' :: "('k \<Rightarrow> 'm) \<times> ('n \<Rightarrow> 'l)" assumes "wiring \<I> \<I>' conv1 fg"   and "wiring \<I>' \<I>'' conv2 fg'" shows "wiring \<I> \<I>'' (conv1 \<odot> conv2) (fg \<circ>\<^sub>w fg')"
uvs_2: fixes u :: "'a"   and v :: "'c" assumes "(u, v) \<in> H ` ball (x, y) r" shows "u = fst (G (u, v))"
left_translations_of_monoid_axioms: shows "left_translations_of_monoid M (\<cdot>) \<one>"
Span_subgroup_props_2: fixes Us :: "'a list" assumes "set Us \<subseteq> carrier R" shows "\<zero> \<in> Span K Us"
Reachable_zero: fixes k :: "nat"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and s :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int" assumes "k = 0"   and "t = s" shows "(M, l, s, k, t) \<in> Reachable"
bind_lift: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'m1"   and g :: "'a \<Rightarrow> 'm1" shows "bind1 (m1.lift f x) g = bind1 x (g \<circ> f)"
eval_or: fixes a :: "atom fm"   and b :: "atom fm"   and L :: "real list" shows "eval (or a b) L = (eval a L \<or> eval b L)"
f2: fixes gpv :: "('g, 'c, 'e) gpv"   and gpv' :: "('h, 'd, 'f) gpv"   and x :: "('g, 'c, ('g, 'c, 'e) rpv) generat"   and y :: "('h, 'd, ('h, 'd, 'f) rpv) generat"   and a :: "('g, 'c, 'e) rpv"   and b :: "('h, 'd, 'f) rpv"   and a' :: "'e"   and \<alpha> :: "'f"   and \<beta> :: "'e"   and b' :: "'f" assumes "local.foo outs_gpv outs_gpv gpv gpv' C C'"   and "x \<in> set_spmf (the_gpv gpv)"   and "y \<in> set_spmf (the_gpv gpv')"   and "a \<in> generat_conts x"   and "b \<in> generat_conts y"   and "R' a' \<alpha>"   and "R' \<beta> b'" shows "local.foo outs_gpv outs_gpv (a a') (b b') C C'"
fold_insert: fixes B :: "'a set"   and x :: "'a"   and z :: "'a" assumes "finite B"   and "x \<notin> B"   and "insert x B \<subseteq> A"   and "z \<in> A" shows "Finite_Set.fold (\<lambda>y. f y ^^ count M y) z (insert x B) = (f x ^^ count M x) (Finite_Set.fold (\<lambda>y. f y ^^ count M y) z B)"
targetsD_2: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b"   and \<mu> :: "'b" assumes "weak_composition V H"   and "b \<in> weak_composition.targets V H \<mu>" shows "weak_composition.weak_unit V H b"
hom_minus: fixes x :: "'pq mod_ring poly"   and y :: "'pq mod_ring poly" shows "#(x - y) = #x - #y"
division_of_tagged_division: fixes s :: "('b \<times> 'b set) set"   and i :: "'b set" assumes "s tagged_division_of i" shows "snd ` s division_of i"
inf_mono: fixes a :: "'a"   and c :: "'a"   and b :: "'a"   and d :: "'a" assumes "a \<le> c"   and "b \<le> d" shows "a \<sqinter> b \<le> c \<sqinter> d"
subseq_to_zero: fixes w :: "nat \<Rightarrow> 'b"   and i :: "nat" shows "w [i \<rightarrow> 0] = []"
GB_imp_zero_reducibility: fixes G :: "('a \<Rightarrow>\<^sub>0 'c) set"   and f :: "'a \<Rightarrow>\<^sub>0 'c" assumes "punit.is_Groebner_basis G"   and "f \<in> local.punit.pmdl G" shows "(punit.red G)\<^sup>*\<^sup>* f 0"
Abs_freelist_append_append: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" shows "Abs_freelist (xs @ ys @ zs) = Abs_freelist xs + Abs_freelist ys + Abs_freelist zs"
llist_all2_conv_all_lnth: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and xs :: "'e llist"   and ys :: "'f llist" shows "llist_all2 P xs ys = (llength xs = llength ys \<and> (\<forall>n. enat n < llength ys \<longrightarrow> P (lnth xs n) (lnth ys n)))"
continuous_logarithm_on_contractible: fixes S :: "'b set"   and f :: "'b \<Rightarrow> complex" assumes "continuous_on S f"   and "contractible S"   and "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0" obtains g :: "'b \<Rightarrow> complex" where "continuous_on S g" and "\<And>x. x \<in> S \<Longrightarrow> f x = exp (g x)"
Sup_ereal_mult_right': fixes Y :: "'a set"   and x :: "real"   and f :: "'a \<Rightarrow> ereal" assumes "Y \<noteq> {}"   and "0 \<le> x" shows "Sup (f ` Y) * ereal x = (SUP i\<in>Y. f i * ereal x)"
d2_def: shows "a.d2 = prod_list a.dims2"
exists_isOmax: fixes R :: "('c \<times> 'c) set set" assumes "finite R"   and "R \<noteq> {}"   and "\<forall>r\<in>R. Well_order r" shows "\<exists>r. isOmax R r"
web_flowD_KIR: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "web_flow \<Gamma> f"   and "x \<notin> A \<Gamma>"   and "x \<notin> B \<Gamma>" shows "KIR f x"
lossless_oracle: fixes \<sigma> :: "'c \<Rightarrow> bool list option"   and x :: "'c" shows "lossless_spmf (hash.oracle \<sigma> x)"
inverse_functors_induce_meta_adjunction: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'d \<Rightarrow> 'c"   and G :: "'c \<Rightarrow> 'd" assumes "inverse_functors C D F G" shows "meta_adjunction C D F G (\<lambda>x. G) (\<lambda>y. F)"
1: shows "w\<^sup>T * top \<sqinter> e\<^sup>T * top = bot"
same_append_eq: fixes xs :: "'g list"   and ys :: "'g list"   and zs :: "'g list" shows "(xs @ ys = xs @ zs) = (ys = zs)"
parts_agent: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "n \<notin> bad_agent" shows "Agent n \<notin> parts (used s)"
map_key_PP_ideal: fixes F :: "(('a, 'b) pp \<Rightarrow>\<^sub>0 'c) set" shows "Poly_Mapping.map_key PP ` ideal F = ideal (Poly_Mapping.map_key PP ` F)"
mkCone_def: fixes p0 :: "'a"   and p1 :: "'a" shows "a0xa1.mkCone p0 p1 \<equiv> \<lambda>j. if j = X.MkIde False then p0 else if j = X.MkIde True then p1 else null"
cmp_def: shows "TfHgh.cmp = TfHgh_HfHgh.chine"
eucl_sup: fixes x :: "'a"   and y :: "'a" shows "sup x y = (\<Sum>i\<in>Basis. sup (x \<bullet> i) (y \<bullet> i) *\<^sub>R i)"
WTrtFAccNT: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e :: "expr"   and F :: "char list"   and Cs :: "char list list"   and T :: "ty" assumes "P,E,h \<turnstile> e : NT" shows "P,E,h \<turnstile> e\<bullet>F{Cs} : T"
inducts: fixes biG :: "('a, 'b) pre_digraph \<times> ('b \<Rightarrow> 'b)"   and x :: "('a, 'b) pre_digraph \<times> ('b \<Rightarrow> 'b)"   and P :: "('a, 'b) pre_digraph \<times> ('b \<Rightarrow> 'b) \<Rightarrow> bool" assumes "subdivision biG x"   and "bidirected_digraph (fst biG) (snd biG) \<Longrightarrow> P biG"   and "\<And>biI biH u v w uv uw vw. \<lbrakk>subdivision biG biI; P biI; subdivision_step (fst biI) (snd biI) (fst biH) (snd biH) (u, v, w) (uv, uw, vw)\<rbrakk> \<Longrightarrow> P biH" shows "P x"
insort_remove1: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and xs :: "'a list" assumes "class.linorder less_eq less"   and "a \<in> set xs"   and "linorder.sorted less_eq xs" shows "linorder.insort_key less_eq (\<lambda>x. x) a (remove1 a xs) = xs"
IsUniqueIteratorTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and src :: "'a expr"   and k :: "call_kind"   and its :: "String.literal list"   and its_ty :: "'a OCL_Types.type option"   and body :: "'a expr"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E IsUniqueIteratorCall src k its its_ty body : \<tau>" obtains \<tau>' :: "'a OCL_Types.type" and \<sigma> :: "'a OCL_Types.type" and \<rho> :: "'a OCL_Types.type" where "k = ArrowCall" and "\<tau> = Boolean[1]" and "\<Gamma> \<turnstile>\<^sub>I (src, its, its_ty, body) : (\<tau>', \<sigma>, \<rho>)" and "length its \<le> Suc 0"
pi_it_to_it: fixes I :: "('b list \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'b list \<Rightarrow> 'b list) \<Rightarrow> 'b list \<Rightarrow> 'b list" shows "proper_it (it_to_it I) (it_to_it I)"
distinct_5: fixes x2 :: "'b" shows "extC.Val x2 \<noteq> extC.Top"
wf_below_mult_local: fixes r :: "('a \<times> 'a) set"   and X :: "'a multiset" assumes "wf_below r (set_mset X)" shows "wf_below (mult r) {X}"
Suc_Rep_inject': fixes x :: "ind"   and y :: "ind" shows "(Suc_Rep x = Suc_Rep y) = (x = y)"
to_bl_use_of_bl: fixes w :: "'b word"   and bl :: "bool list" shows "(to_bl w = bl) = (w = of_bl bl \<and> length bl = length (to_bl w))"
IH: fixes S :: "(char list \<Rightarrow> int) set"   and S' :: "'av Abs_State.st option"   and ca :: "'av Abs_State.st option acom" assumes "S \<subseteq> \<gamma>\<^sub>o S'"   and "cs1 \<le> \<gamma>\<^sub>c ca" shows "step S cs1 \<le> \<gamma>\<^sub>c (step' S' ca)"
summable_exp: fixes x :: "'b" shows "summable (\<lambda>n. inverse (fact n) * x ^ n)"
S5: shows "(x * (x + 1) + 2 * (x + 1)) div 2 = x + 1 + x * (x + 1) div 2"
mkCone_cone: fixes a :: "'a"   and \<chi> :: "(bool, bool list) concrete_category.arr \<Rightarrow> 'a" assumes "D.cone a \<chi>" shows "D.mkCone (\<chi> J.Zero) = \<chi>"
mkNode_def: fixes c :: "'e"   and l :: "(('c \<times> 'd) \<times> 'e \<times> 'c \<times> 'd) tree"   and a :: "'c \<times> 'd"   and r :: "(('c \<times> 'd) \<times> 'e \<times> 'c \<times> 'd) tree" shows "mkNode c l a r \<equiv> \<langle>l, (a, c, min_kp a l r), r\<rangle>"
eqd_equal_suf: fixes x :: "'b list"   and y :: "'b list"   and u :: "'b list"   and v :: "'b list" assumes "x \<cdot> y = u \<cdot> v"   and "\<^bold>|y\<^bold>| = \<^bold>|v\<^bold>|" shows "x = u \<and> y = v"
lunit'_simps_1: fixes f :: "'b" assumes "D.ide f" shows "D.arr \<l>\<^sub>D\<^sup>-\<^sup>1[f]"
R: shows "\<forall>x\<in>S. \<exists>e>0. closed (cball x e \<inter> S)"
simps_1: fixes k :: "nat"   and x :: "'a" shows "ins k x Leaf = Up\<^sub>i Leaf x Leaf"
thetaZOSeqD_ZObis: shows "thetaZOSeqD \<subseteq> ZObis"
boundOutputChainEq: fixes xvec :: "name list"   and B :: "('d, 'e, 'f) boundOutput"   and yvec :: "name list"   and B' :: "('d, 'e, 'f) boundOutput" assumes "\<lparr>\<nu>*xvec\<rparr>B = \<lparr>\<nu>*yvec\<rparr>B'"   and "xvec \<sharp>* yvec"   and "length xvec = length yvec" shows "\<exists>p. set p \<subseteq> set xvec \<times> set yvec \<and> distinctPerm p \<and> B = p \<bullet> B' \<and> set (map fst p) \<subseteq> supp B \<and> xvec \<sharp>* B' \<and> yvec \<sharp>* B"
ro: shows "relation_order (red F) (\<prec>\<^sub>p) (dgrad_p_set d m)"
region_integral_is_one_twoD_integral: shows "LBINT x. LBINT y. f (y, x) = integral\<^sup>L (lborel \<Otimes>\<^sub>M lborel) f"
Abs_Store_inject: fixes x :: "StoreImpl"   and y :: "StoreImpl" assumes "x \<in> Store"   and "y \<in> Store" shows "(Abs_Store x = Abs_Store y) = (x = y)"
convex_def: fixes s :: "'e set" shows "convex s = (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)"
h7: assumes "Av < 0"   and "0 < Bv" shows "(Av + Bv * sqrt Cv < 0) = (Av < 0 \<and> Bv\<^sup>2 * Cv < Av\<^sup>2 \<or> Bv \<le> 0 \<and> (Av < 0 \<or> Av\<^sup>2 < Bv\<^sup>2 * Cv))"
leg1_composite_1: shows "H\<nu>H\<pi>\<rho>.leg1 = \<nu>.leg1 \<cdot> \<nu>\<pi>\<rho>.Prj\<^sub>1"
assms_1: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set" assumes "X \<in> set XL" shows "good X"
src_src: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "src (src \<mu>) = src \<mu>"