hcju/isabelleps huggingface dataset:
real_transfer_operator_foT_g: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "integrable M (\<lambda>x. f (T x) * g x)" shows "AE x in M. real_transfer_operator (\<lambda>x. f (T x) * g x) x = f x * real_transfer_operator g x"
self_codomain_iff_ide: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "partial_magma C" shows "(a \<in> partial_magma.codomains C a) = partial_magma.ide C a"
normalize_nnf_pos: fixes m :: "'a common_primitive match_expr"   and m' :: "'a common_primitive match_expr" assumes "normalized_nnf_match m"   and "\<not> has_disc_negated disc False m"   and "m' \<in> set (normalize_pos m)" shows "normalized_nnf_match m'"
assms: shows "mtype(CT,m,C) = Cs \<rightarrow> C0"
intro: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs heap_is_wellformed parent_child_rel get_disconnected_nodes"   and "l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs heap_is_wellformed parent_child_rel" shows "l_get_parent_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs heap_is_wellformed parent_child_rel get_disconnected_nodes get_disconnected_nodes_locs"
upto_enum_red': fixes X :: "'b word" assumes "1 \<le> X" shows "[0 .e. X - 1] = map word_of_nat [0..<unat X]"
simps_1: fixes a :: "atomUni" shows "negUni (AtomUni a) = AtomUni (aNegUni a)"
ereal_0_le_uminus_iff: fixes a :: "ereal" shows "(0 \<le> - a) = (a \<le> 0)"
with_half_stepsizes_def: fixes x :: "'d set" shows "with_half_stepsizes x = x"
consistent_signs_at_roots_eq: fixes p :: "real poly"   and qs :: "real poly list" assumes "p \<noteq> 0" shows "consistent_signs_at_roots p qs = set (characterize_consistent_signs_at_roots p qs)"
combine_hom: fixes Ks :: "'a list"   and Us :: "'a list" assumes "set Ks \<subseteq> carrier R"   and "set Us \<subseteq> carrier R" shows "S.combine (map h Ks) (map h Us) = h (R.combine Ks Us)"
iterate_add_def: fixes n :: "nat"   and a :: "'b" shows "iterate_add n a = ((+) a ^^ n) (0::'b)"
cod_simp: fixes f :: "'a \<times> 'a" assumes "VoV.arr f" shows "VoV.cod f = VxV.cod f"
oalist_inv_alt: fixes xs :: "('a \<times> 'b) list"   and ko :: "'a key_order" shows "ko.oalist_inv (xs, ko) = ko.oalist_inv_raw ko xs"
RedBinOp: fixes bop :: "bop"   and v\<^sub>1 :: "val"   and v\<^sub>2 :: "val"   and v :: "val"   and E :: "char list \<Rightarrow> ty option"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and 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" assumes "binop (bop, v\<^sub>1, v\<^sub>2) = \<lfloor>v\<rfloor>" shows "P,E \<turnstile> \<langle>Val v\<^sub>1 \<guillemotleft>bop\<guillemotright> Val v\<^sub>2,s\<rangle> \<rightarrow> \<langle>Val v,s\<rangle>"
Throw: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and a :: "nat"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>throw e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Throw a,s\<^sub>1\<rangle>"
card_Plus: fixes A :: "'a set"   and B :: "'b set" assumes "finite A"   and "finite B" shows "card (A <+> B) = card A + card B"
assms_3: assumes "nsqn\<^sub>r r = nsqn\<^sub>r r' \<and> \<pi>\<^sub>5 r' \<le> \<pi>\<^sub>5 r" shows "P r r'"
1: shows "\<forall>xa\<in>{1..m}. \<forall>y\<in>{1..m}. xa \<noteq> y \<longrightarrow> (A(x := A x - {Suc j})) xa \<inter> (A(x := A x - {Suc j})) y = {}"
bisimE_4: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<sim> Q" shows "\<Psi> \<rhd> Q \<sim> P"
alphaAbs_all_distinct_qAFresh_def: fixes xs :: "'f"   and x :: "'g"   and X :: "('h, 'i, 'f, 'g, 'j) qTerm"   and xs' :: "'f"   and x' :: "'g"   and X' :: "('h, 'i, 'f, 'g, 'j) qTerm" shows "alphaAbs_all_distinct_qAFresh xs x X xs' x' X' \<equiv> xs = xs' \<and> (\<forall>y. y \<notin> {x, x'} \<and> qAFresh xs y X \<and> qAFresh xs y X' \<longrightarrow> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
cong_mult_lcancel_nat: fixes k :: "nat"   and m :: "nat"   and a :: "nat"   and b :: "nat" assumes "coprime k m" shows "[k * a = k * b] (mod m) = [a = b] (mod m)"
sets_into_space: fixes x :: "'b set"   and M :: "'b measure" assumes "x \<in> sets M" shows "x \<subseteq> space M"
mk_disjoint_finsert: fixes a :: "'b"   and A :: "'b fset" assumes "a |\<in>| A" shows "\<exists>B. A = finsert a B \<and> a |\<notin>| B"
distinct_14: fixes x51 :: "bexp"   and x52 :: "com"   and x21 :: "char list"   and x22 :: "aexp" shows "WHILE x51 DO x52 \<noteq> x21 ::= x22"
ipresAbs_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('e, 'f, 'varSort, 'g, 'h, 'var, 'c, 'd, 'i) model_scheme" shows "ipresAbs h hA MOD \<equiv> \<forall>xs x s X. isInBar (xs, s) \<and> wls s X \<longrightarrow> hA (Abs xs x X) = igAbs MOD xs x (h X)"
euler_genus_def: shows "G'.euler_genus \<equiv> (int (2 * card sccs) - int (card isolated_verts) - G'.euler_char) div 2"
mQmp_def: fixes M :: "('e, 'f, 'g) Module_scheme"   and H :: "'e set"   and N :: "'e set" shows "Mp\<^bsub>M  H,N\<^esub> = (\<lambda>X\<in>set_mr_cos M H. {z. \<exists>x\<in>X. \<exists>y\<in>N. y \<plusminus>\<^bsub>M\<^esub> x = z})"
b_def: shows "b = (if 0 < r then a else a + pi)"
proper_design_axioms: shows "proper_design \<V>' \<B>'"
weak_unit_self_composable_3: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "weak_composition V H"   and "weak_composition.weak_unit V H a" shows "H a a \<noteq> partial_magma.null V"
LSC_sim_Resolution1: fixes S :: "'b formula"   and T :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and k :: "'b" assumes "S \<^bold>\<or> T, \<Gamma> \<Rightarrow>\<^sub>n" shows "Atom k \<^bold>\<or> S, \<^bold>\<not> (Atom k) \<^bold>\<or> T, \<Gamma> \<Rightarrow>\<^sub>n"
assms_4: fixes x :: "real" assumes "1 \<le> x" shows "0 < h x"
\<chi>\<^sub>1_def: shows "\<chi>\<^sub>1 \<equiv> principal_dchar n"
d: shows "content (cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) < e"
apartment_chamber_system_def: fixes X :: "'a set set" assumes "X \<in> \<A>" shows "ChamberComplex.chamber_system X \<equiv> {C. SimplicialComplex.maxsimp X C}"
dmfn0: fixes n :: "nat" shows "f n - f 0 \<in> carrier_mat d d"
C1_ket: fixes x :: "'a" shows "C1 (ket x) = (case \<pi> x of None \<Rightarrow> 0 | Some i \<Rightarrow> ket i)"
assms: shows "A \<subseteq>\<^sub>\<circ> B"
a_rcosets_inv_mult_group_eq: fixes M :: "'a set" assumes "M \<in> a_rcosets H" shows "a_set_inv M <+> M = H"
sp_in_extends_imp2: fixes g :: "('d \<times> nat, 'e) rgraph_scheme"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set"   and v :: "'d \<times> nat"   and v' :: "'d \<times> nat"   and subs' :: "(('d \<times> nat) \<times> 'd \<times> nat) set"   and v1 :: "'d \<times> nat"   and es :: "('d \<times> nat) edge list"   and v2 :: "'d \<times> nat" assumes "wf_sub_rel_of g subs"   and "SubExt.extends g subs (v, v') subs'"   and "SubRel.subpath g v1 es v2 subs'"   and "v1 \<noteq> v" shows "SubRel.subpath g v1 es v2 subs \<or> (\<exists>es1 es2. es = es1 @ es2 \<and> es1 \<noteq> [] \<and> SubRel.subpath g v1 es1 v subs \<and> SubRel.subpath g v es2 v2 subs')"
inorder_delete: fixes t :: "(('c \<times> 'd) \<times> nat) tree"   and x :: "'c" assumes "sorted1 (Tree2.inorder t)" shows "Tree2.inorder (AVL_Map.delete x t) = AList_Upd_Del.del_list x (Tree2.inorder t)"
runit_hcomp_3: fixes f :: "'b"   and g :: "'b" assumes "D.ide f"   and "D.ide g"   and "src\<^sub>D f = trg\<^sub>D g" shows "\<r>\<^sub>D[f \<star>\<^sub>D g] \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[f, g, src\<^sub>D g] = f \<star>\<^sub>D \<r>\<^sub>D[g]"
fps_of_ratfps_0: shows "fps_of_ratfps 0 = 0"
zero_left: fixes prod :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and b :: "'e" assumes "bounded_bilinear prod" shows "prod (0::'d) b = (0::'f)"
converseE: fixes yx :: "'a \<times> 'b"   and r :: "('b \<times> 'a) set" assumes "yx \<in> r\<inverse>" obtains x :: "'b" and y :: "'a" where "yx = (y, x)" and "(x, y) \<in> r"
h'h: fixes x :: "'b" assumes "x \<in> S'" shows "h' x = h x"
isArrLenLoc_simps_1: fixes cf :: "CAttId"   and a :: "nat" shows "isArrLenLoc (objLoc cf a) = False"
not_choice_simps_1: fixes l :: "'d"   and fg :: "'a \<Rightarrow> 'a set"   and p :: "('a, 'b, 'c, 'd) seqp" shows "not_choice ({l}\<langle>fg\<rangle> p)"
set_restriction_absorb: fixes f :: "'a set \<Rightarrow> 'a set"   and A :: "'a set" assumes "set_restriction f" shows "f (f A) = f A"
set_enforce_spmf: fixes P :: "'d \<Rightarrow> bool"   and p :: "'d spmf" shows "set_spmf (enforce_spmf P p) = {a \<in> set_spmf p. P a}"
eq_imp_le: fixes m :: "nat"   and n :: "nat" assumes "m = n" shows "m \<le> n"
up_filter_dense_antitone_dense: fixes x :: "'c"   and y :: "'c"   and z :: "'c" shows "dense (x \<squnion> - x \<squnion> y) \<and> dense (x \<squnion> - x \<squnion> y \<squnion> z)"
order_iff: fixes a :: "'b::semilattice_inf"   and b :: "'b::semilattice_inf" shows "((a::'b::semilattice_inf) \<le> (b::'b::semilattice_inf)) = (a = a \<sqinter> b)"
prod_dfs_start: fixes i :: "nat"   and j :: "nat" assumes "dfa_is_node A i"   and "dfa_is_node B j" shows "fst (prod_dfs A B (i, j)) ! i ! j = Some 0"
norm_iff: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" shows "((\<lambda>x. norm (f x)) \<in> \<Theta>[F](\<lambda>x. norm (g x))) = (f \<in> \<Theta>[F](g))"
admissible\<^sub>1: fixes f :: "face" assumes "f \<in> \<F> g" shows "\<d> |vertices f| \<le> w f"
subst_apply_pair_pair: fixes \<I> :: "'f \<Rightarrow> ('d, 'e) Term.term"   and t :: "('d, 'f) Term.term"   and s :: "('d, 'f) Term.term" shows "(t, s) \<cdot>\<^sub>p \<I> = (t \<cdot> \<I>, s \<cdot> \<I>)"
ASSIGN: fixes e :: "Expr"   and G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and x :: "Var" assumes "Expr_low e" shows "G \<rhd> Assign x e : Sec (\<lambda>(s, t, \<beta>). \<exists>r. s = (update (fst r) x (evalE e (fst r)), snd r) \<and> r \<equiv>\<^sub>\<beta> t)"
assms_1: fixes n :: "nat" shows "u n \<in> extended_distance_set"
cSUP_upper2: fixes f :: "'c \<Rightarrow> 'b multiset"   and A :: "'c set"   and x :: "'c"   and u :: "'b multiset" assumes "subset_mset.bdd_above (f ` A)"   and "x \<in> A"   and "u \<subseteq># f x" shows "u \<subseteq># Sup (f ` A)"
monoid_mult_hom_1_axioms: shows "monoid_mult_hom_1 hom"
intros_1: fixes f :: "'a \<Rightarrow> 'a"   and s :: "'a" shows "Sem (Basic f) s (f s)"
le_bot: fixes a :: "'a" assumes "a \<le> bot" shows "a = bot"
retraction_subset: fixes S :: "'a set"   and T :: "'a set"   and r :: "'a \<Rightarrow> 'a"   and s' :: "'a set" assumes "retraction S T r"   and "T \<subseteq> s'"   and "s' \<subseteq> S" shows "retraction s' T r"
ConjunctiveD: fixes a :: "'c"   and Inf_b :: "'d set \<Rightarrow> 'd"   and Inf_c :: "'e set \<Rightarrow> 'e"   and times_abc :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and A :: "'d set" assumes "a \<in> Conjunctive.Conjunctive Inf_b Inf_c times_abc" shows "times_abc a (Inf_b A) = Inf_c (times_abc a ` A)"
simps_1: fixes x21 :: "char list"   and x22 :: "aexp"   and y21 :: "char list"   and y22 :: "aexp" shows "(x21 ::= x22 = y21 ::= y22) = (x21 = y21 \<and> x22 = y22)"
minus_mod_self1: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and uminus :: "'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus" shows "modulo (minus b a) b = modulo (uminus a) b"
divide_right_iff: fixes h :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<forall>\<^sub>F x in F. h x \<noteq> (0::'b)" shows "((\<lambda>x. f x / h x) \<in> L F (\<lambda>x. g x / h x)) = (f \<in> L F g)"
erf_remainder'_def: fixes n :: "nat"   and x :: "real" shows "erf_remainder' n x = exp (x\<^sup>2) * sqrt pi * erf_remainder n x"
good_def: fixes v :: "'a" shows "good v = {\<sigma>. strategy p \<sigma> \<and> winning_strategy p \<sigma> v}"
option_while_def: fixes C :: "'b \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'b option"   and r :: "'b" shows "option_while C B r \<equiv> if \<exists>s. (Some r, s) \<in> option_while' C B then THE s. (Some r, s) \<in> option_while' C B else None"
HMA_diagonal_step: fixes i :: "nat"   and j :: "nat" assumes "i < min CARD('b) CARD('c)"   and "j < min CARD('b) CARD('c)" shows "(Mod_Type_Connect.HMA_M ===> (=) ===> (=) ===> Mod_Type_Connect.HMA_M) (\<lambda>A. diagonal_step_JNF A i j) (\<lambda>B. diagonal_step B i j)"
map_restrict_dom_compl: fixes f :: "'c \<Rightarrow> 'd option" shows "f |` (- dom f) = Map.empty"
mem_Times_mset_iff: fixes x :: "'a \<times> 'b"   and A :: "'a multiset"   and B :: "'b multiset" shows "(x \<in># A \<times># B) = (fst x \<in># A \<and> snd x \<in># B)"
float_plus_up_le: fixes x :: "real"   and xa :: "float"   and y :: "float"   and prec :: "nat" assumes "x \<le> real_of_float xa + real_of_float y" shows "x \<le> real_of_float (float_plus_up prec xa y)"
ring_primeE_3: fixes p :: "'a list" assumes "p \<in> carrier (K [X])"   and "pprime K p" shows "prime (K [X]) p"
assms: shows "ccpo.admissible Sup (\<le>) (\<lambda>y. \<not> x \<le> y)"
tllist_ord_TCons_TNil: fixes b :: "'f"   and x :: "'g"   and xs :: "('g, 'f) tllist"   and b2 :: "'f" shows "tllist_ord b (TCons x xs) (TNil b2) = False"
pointsto_def: fixes v :: "char list"   and n :: "int" shows "v \<hookrightarrow> n = (\<lambda>(s, c). s = [v \<mapsto> n] \<and> c = 0)"
assms_5: shows "ptr \<notin> set |h \<turnstile> get_dom_component (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^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 document_ptr)|\<^sub>r"
add_mset_add_single: fixes a :: "'a"   and A :: "'a multiset" shows "add_mset a A = A \<oplus> a"
dist_le_1_imp_domain_eq: fixes P :: "'d \<Rightarrow>\<^sub>F 'e"   and Q :: "'d \<Rightarrow>\<^sub>F 'e" assumes "dist P Q < 1" shows "domain P = domain Q"
I_subst_peps2: fixes f :: "atom fm"   and r :: "real"   and cs :: "real list"   and xs :: "real list"   and x :: "real" assumes "nqfree f"   and "r + \<langle>cs,xs\<rangle> < x"   and "nolb f xs (r + \<langle>cs,xs\<rangle>) x"   and "\<forall>y\<in>{r + \<langle>cs,xs\<rangle><..x}. R.I f (y # xs) \<and> y \<notin> EQ2 f xs" shows "R.I (subst\<^sub>+ f (r, cs)) xs"
dvd_add_triv_right_iff: 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 a :: "'c"   and b :: "'c" assumes "class.comm_semiring_1_cancel plus minus zero times one" shows "dvd.dvd times a (plus b a) = dvd.dvd times a b"
findMin_def: fixes bq :: "('a, 'b) BinomialHeapStruc.BinomialTree list" shows "BinomialHeapStruc.findMin bq = (let min = BinomialHeapStruc.getMinTree bq in (BinomialHeapStruc.val min, BinomialHeapStruc.prio min))"
after_summary_empty_summary: fixes M :: "'t zmultiset" shows "after_summary M {}\<^sub>A = {#}\<^sub>z"
mem_case_prodE: fixes z :: "'b"   and c :: "'c \<Rightarrow> 'd \<Rightarrow> 'b set"   and p :: "'c \<times> 'd" assumes "z \<in> (case p of (x, xa) \<Rightarrow> c x xa)" obtains x :: "'c" and y :: "'d" where "p = (x, y)" and "z \<in> c x y"
pwd_spy_ii_unused: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "\<exists>C. PriKey C \<notin> used (fst (pwd_spy_ii n))"
special_soundness_def: shows "\<Sigma>1.special_soundness \<equiv> \<forall>h e e' a z z'. h \<in> valid_pub1 \<longrightarrow> e \<in> carrier L \<longrightarrow> e' \<in> carrier L \<longrightarrow> e \<noteq> e' \<longrightarrow> check1 h a e z \<longrightarrow> check1 h a e' z' \<longrightarrow> lossless_spmf (\<A>ss1 h (a, e, z) (a, e', z')) \<and> (\<forall>w'\<in>set_spmf (\<A>ss1 h (a, e, z) (a, e', z')). (h, w') \<in> Rel1)"
1: shows "(f ` U \<subseteq> topspace Y) = (U \<subseteq> topspace X)"
H: fixes Aa :: "'b multiset"   and j :: "'a" assumes "\<forall>i. i \<in># dom_m B \<longrightarrow> (\<forall>j. (j, i) \<in> K \<longrightarrow> (the (fmlookup A j), the (fmlookup B i)) \<in> R)"   and "remove1_mset p' (dom_m B) = add_mset p' Aa"   and "(j, p') \<in> K" shows "False"
preserves_arr: fixes f :: "'a" assumes "A.arr f" shows "B.arr (G f)"
single_diff: fixes k :: "'b"   and a :: "'c"   and b :: "'c" shows "Poly_Mapping.single k (a - b) = Poly_Mapping.single k a - Poly_Mapping.single k b"
get_shadow_root_locs_def: shows "get_shadow_root_locs \<equiv> local.a_get_shadow_root_locs"
pop_def: fixes PDPE :: "'a set list \<times> 'a set \<times> ('a \<times> 'a) set" shows "pop PDPE \<equiv> let (p, D, pE) = PDPE; (p, V) = (butlast p, last p); D = V \<union> D in (p, D, pE)"
evalOP_Fix: fixes P :: "db"   and V :: "db" assumes "P<DBFix P/0> \<Down> V" shows "DBFix P \<Down> V"
range_varsE: fixes x :: "'c"   and \<sigma> :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "x \<in> range_vars \<sigma>" obtains t :: "('d, 'c) Term.term" where "x \<in> fv t" and "t \<in> subst_range \<sigma>"
set_filter: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "set (filter P xs) = {x \<in> set xs. P x}"
comp_def: fixes g :: "'a1 \<times> 'a2"   and f :: "'a1 \<times> 'a2" shows "g \<cdot> f = (if (C1.arr (fst f) \<and> C2.arr (snd f)) \<and> (C1.arr (fst g) \<and> C2.arr (snd g)) \<and> (if C1.arr (fst f) \<and> C2.arr (snd f) then (C1.cod (fst f), C2.cod (snd f)) else (C1.null, C2.null)) = (if C1.arr (fst g) \<and> C2.arr (snd g) then (C1.dom (fst g), C2.dom (snd g)) else (C1.null, C2.null)) then (fst g \<cdot>\<^sub>1 fst f, snd g \<cdot>\<^sub>2 snd f) else (C1.null, C2.null))"
induced_automorphism_def: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set"   and v :: "'b" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "OpposedThinChamberComplexFoldings.induced_automorphism X f g v \<equiv> if v \<in> f ` \<Union> X then g v else restrict1 f (g ` \<Union> X) v"
Xm1: shows "\<forall>y\<in>X. y $ m = (1::'a)"
subtensor_prod_with_vec: fixes A :: "'b tensor"   and i :: "nat"   and B :: "'b tensor" assumes "order A = 1"   and "i < hd (dims A)" shows "subtensor (A \<otimes> B) i = lookup A [i] \<cdot> B"
coeffs_def: fixes p :: "'b poly" shows "coeffs p = (if p = 0 then [] else map (coeff p) [0..<Suc (degree p)])"
nat_distrib_1: fixes m :: "nat"   and n :: "nat"   and k :: "nat" shows "(m + n) * k = m * k + n * k"
bijI': fixes f :: "'c \<Rightarrow> 'd" assumes "\<And>x y. (f x = f y) = (x = y)"   and "\<And>y. \<exists>x. y = f x" shows "bij f"
ln_mono_strict: fixes x :: "real"   and y :: "real" assumes "0 < x"   and "0 < y"   and "x < y" shows "ln x < ln y"
assms_4: shows "\<forall>i. \<not> is_zero_row i A \<longrightarrow> A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a)) \<in> associates"
product_not_empty: fixes xs :: "'b list"   and ys :: "'c list" assumes "xs \<noteq> []"   and "ys \<noteq> []" shows "List.product xs ys \<noteq> []"
tensor_lookup: fixes A :: "'a tensor"   and e :: "nat list \<Rightarrow> 'a" assumes "\<And>is. is \<lhd> dims A \<Longrightarrow> lookup A is = e is" shows "tensor_from_lookup (dims A) e = A"
indep_system_def: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool" shows "indep_system carrier indep \<equiv> (finite carrier \<and> (\<forall>X. indep X \<longrightarrow> X \<subseteq> carrier)) \<and> (\<exists>X. indep X) \<and> (\<forall>X Y. indep X \<longrightarrow> Y \<subseteq> X \<longrightarrow> indep Y)"
P1: shows "Bet B R Q \<and> Cong A P B R"
unitor_coincidence_1: fixes a :: "'a" assumes "B.obj a" shows "B.lunit a = \<i>\<^sub>B[a]"
k_1: shows "5 * delta * real k + 15 / 2 * delta + C / 2 \<le> D"
valuesum_rec: fixes m :: "('c, 'd) mapping" assumes "finite (dom (Mapping.lookup m))" shows "valuesum m = (if Mapping.is_empty m then 0::'d else let l = SOME l. l \<in> Mapping.keys m in the (Mapping.lookup m l) + valuesum (Mapping.delete l m))"
fv\<^sub>s\<^sub>e\<^sub>t_union: fixes M :: "('b, 'a) Term.term set"   and N :: "('b, 'a) Term.term set" shows "fv\<^sub>s\<^sub>e\<^sub>t (M \<union> N) = fv\<^sub>s\<^sub>e\<^sub>t M \<union> fv\<^sub>s\<^sub>e\<^sub>t N"
obspne: shows "obsp \<sigma> i \<noteq> obsp \<sigma>' i'"
newton_poly_def: fixes n :: "nat" shows "local.newton_poly n = local.b 0 n"
restrict_list_singleton: fixes ks :: "'d list"   and f :: "'d \<times> 'd \<Rightarrow> 'c"   and l :: "'d"   and i :: "'d"   and j :: "'d" shows "(ks \<langle>f\<rangle> [l]) (i, j) = (if List.member ks i \<and> j = l then f (i, j) else bot)"
suffix_same_cases: fixes xs\<^sub>1 :: "'c list"   and ys :: "'c list"   and xs\<^sub>2 :: "'c list" assumes "suffix xs\<^sub>1 ys"   and "suffix xs\<^sub>2 ys" shows "suffix xs\<^sub>1 xs\<^sub>2 \<or> suffix xs\<^sub>2 xs\<^sub>1"
wt_eenv: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "Ik.wtE \<xi>" shows "TE.wtE (eenv \<xi>)"
action_dom_def: fixes s1 :: "('a, 'b) fmap"   and s2 :: "('a, 'c) fmap" shows "action_dom s1 s2 \<equiv> fmdom' s1 \<union> fmdom' s2"
assms: shows "Abs x s \<rightarrow>\<^sub>\<beta> t"
OV_inv1_def: shows "OV_inv1 = {s. \<forall>r Q v. quorum_for Q v (votes s r) \<longrightarrow> (\<forall>Q'\<in>Quorum. quorum_for Q' v (obsv_state.obs s r))}"
f_inv_f: fixes x :: "'a" shows "inv f (inv (inv f) x) = x"
coeff_mult_monom: fixes p :: "'a poly"   and a :: "'a"   and d :: "nat"   and i :: "nat" shows "coeff (p * monom a d) i = (if d \<le> i then a * coeff p (i - d) else (0::'a))"
filterlim_cong: fixes F1 :: "'a filter"   and F1' :: "'a filter"   and F2 :: "'b filter"   and F2' :: "'b filter"   and f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a" assumes "F1 = F1'"   and "F2 = F2'"   and "\<forall>\<^sub>F x in F2. f x = g x" shows "filterlim f F1 F2 = filterlim g F1' F2'"
lel_lemma: fixes F :: "'d program"   and A :: "'d set"   and B :: "'d set" assumes "F \<in> A \<longmapsto> B" shows "F \<in> (reachable F \<inter> A) leadsTo[Pow (reachable F)] B"