hcju/isabelleps huggingface dataset:
P2: shows "[p] \<cdot> \<one> [^] nat (ord_Zp x) \<otimes> [p] \<cdot> \<one> [^] nat (ord_Zp y) \<otimes> ac_Zp (x \<otimes> y) = [p] \<cdot> \<one> [^] nat (ord_Zp x) \<otimes> ac_Zp x \<otimes> ([p] \<cdot> \<one> [^] nat (ord_Zp y) \<otimes> ac_Zp y)"
lassocr\<^sub>w_def: shows "lassocr\<^sub>w = (rsuml, lsumr)"
abs_eq: shows "1 = \<omega>\<^sub>z^ 0"
IH_4: shows "((l, insert\<langle>t,s\<rangle>) \<in> set A) = ((l, insert\<langle>t,s\<rangle>) \<in> set (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A))"
is_pot_ord: fixes u :: "'t"   and v :: "'t" assumes "is_pot_ord" shows "(u \<preceq>\<^sub>t v) = (component_of_term u < component_of_term v \<or> component_of_term u = component_of_term v \<and> pp_of_term u \<preceq> pp_of_term v)"
on_curve_def: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and p :: "'b point" assumes "class.ell_field divide inverse times one plus zero minus uminus" shows "ell_field.on_curve times one plus a b p = (case p of Infinity \<Rightarrow> True | Point x y \<Rightarrow> power.power one times y 2 = plus (plus (power.power one times x 3) (times a x)) b)"
residue_minus: fixes k :: "nat"   and x :: "int"   and y :: "int" shows "x \<ominus>\<^bsub>residue_ring (p ^ k)\<^esub> y = (x - y) mod p ^ k"
assms_2: shows "f \<in> F \<rightarrow>\<^sub>M N"
space_as_set_inverse: fixes x :: "'a ccsubspace" shows "Abs_clinear_space (space_as_set x) = x"
l_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf known_ptrs heap_is_wellformed parent_child_rel get_child_nodes get_child_nodes_locs get_disconnected_nodes get_parent get_parent_locs get_ancestors get_ancestors_locs get_root_node get_root_node_locs"
finite_bound: fixes rel :: "(nat \<times> nat) set" assumes "finite rel" shows "\<exists>n. is_bound rel n"
real_cond_exp_discount_stock_q_const: fixes N :: "bool stream measure"   and q :: "real"   and n :: "nat" assumes "N = bernoulli_stream q"   and "q = (1 + r - d) / (u - d)" shows "AEeq N (real_cond_exp N (G n) (discounted_value r (prices Mkt stk) (Suc n))) (discounted_value r (prices Mkt stk) n)"
not_PL: fixes A :: "'a set"   and p :: "'a" shows "(A \<turnstile> .\<not> p) = (A \<union> {p} \<turnstile> \<bottom>)"
assms_4: shows "fst (rep_nat_term u) = 0"
REC_le_RECT: fixes body :: "('d \<Rightarrow> 'c) \<Rightarrow> 'd \<Rightarrow> 'c"   and x :: "'d" shows "REC body x \<le> REC\<^sub>T body x"
alpha_Tree_valid_Tree: fixes t1 :: "('a, 'pred, 'act) Tree"   and t2 :: "('a, 'pred, 'act) Tree"   and P :: "'state" assumes "t1 =\<^sub>\<alpha> t2" shows "valid_Tree P t1 = valid_Tree P t2"
exec_gpv_fixp_parallel_induct: fixes P :: "(('d, 'e, 'f) gpv \<Rightarrow> 'g \<Rightarrow> ('d \<times> 'g) spmf) \<Rightarrow> (('h, 'i, 'j) gpv \<Rightarrow> 'k \<Rightarrow> ('h \<times> 'k) spmf) \<Rightarrow> bool"   and calleea :: "'g \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'g) spmf"   and callee :: "'k \<Rightarrow> 'i \<Rightarrow> ('j \<times> 'k) spmf" assumes "ccpo.admissible (prod_lub spmf.lub_fun spmf.lub_fun) (rel_prod spmf.le_fun spmf.le_fun) (\<lambda>x. P (curry (fst x)) (curry (snd x)))"   and "P (\<lambda>_ y. return_pmf None) (\<lambda>_ y. return_pmf None)"   and "\<And>f g. P f g \<Longrightarrow> P (\<lambda>a aa. the_gpv a \<bind> (\<lambda>a. case a of Pure x \<Rightarrow> return_spmf (x, aa) | IO out c \<Rightarrow> calleea aa out \<bind> (\<lambda>a. case a of (x, y) \<Rightarrow> f (c x) y))) (\<lambda>a aa. the_gpv a \<bind> (\<lambda>a. case a of Pure x \<Rightarrow> return_spmf (x, aa) | IO out c \<Rightarrow> callee aa out \<bind> (\<lambda>a. case a of (x, y) \<Rightarrow> g (c x) y)))" shows "P (exec_gpv calleea) (exec_gpv callee)"
uedge_def: shows "uedge \<equiv> \<lambda>(a, b). {a, b}"
assms_2: shows "z \<equiv> \<lambda>n. (any, replicate n False)"
leg_simps_1: shows "C.arr trg_\<mu>.cod.leg0"
finite_enumerate_Ex: fixes S :: "'b set"   and s :: "'b" assumes "finite S"   and "s \<in> S" shows "\<exists>n<card S. wellorder_class.enumerate S n = s"
has_lock_has_locks_Suc: fixes l :: "('b \<times> nat) option"   and t :: "'b" assumes "has_lock l t" shows "\<exists>m. has_locks l t = Suc m"
WCDNonInterferenceGraph: fixes prog :: "cmd"   and nx :: "w_node" assumes "While_CFGExit.inner_node prog nx" shows "NonInterferenceIntraGraph src trg knd (lift_valid_edge (valid_edge prog) sourcenode targetnode kind (_Entry_) (_Exit_)) NewEntry (lift_Def (Defs prog) (_Entry_) (_Exit_) H L) (lift_Use (Uses prog) (_Entry_) (_Exit_) H L) id (PDG.PDG_BS src trg (lift_valid_edge (valid_edge prog) sourcenode targetnode kind (_Entry_) (_Exit_)) (lift_Def (Defs prog) (_Entry_) (_Exit_) H L) (lift_Use (Uses prog) (_Entry_) (_Exit_) H L) (StrongPostdomination.weak_control_dependence src trg (lift_valid_edge (valid_edge prog) sourcenode targetnode kind (_Entry_) (_Exit_)) NewExit)) NewExit H L (LDCFG_node.Node (_Entry_)) (LDCFG_node.Node (_Exit_))"
irrefl: fixes U :: "'a set"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ls :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "order_ow U le ls"   and "a \<in> U" shows "\<not> ls a a"
discounted_mult: fixes r :: "real"   and X :: "nat \<Rightarrow> 'b \<Rightarrow> real"   and Y :: "nat \<Rightarrow> 'b \<Rightarrow> real" shows "\<forall>n w. discounted_value r (\<lambda>m x. X m x * Y m x) n w = X n w * discounted_value r Y n w"
sams_lea2_suma2__conga123: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p"   and D' :: "'p"   and E' :: "'p"   and F' :: "'p"   and G :: "'p"   and H :: "'p"   and I :: "'p" assumes "A B C LeA A' B' C'"   and "D E F LeA D' E' F'"   and "SAMS A' B' C' D' E' F'"   and "A B C D E F SumA G H I"   and "A' B' C' D' E' F' SumA G H I" shows "A B C CongA A' B' C'"
suminf_exist_split: fixes r :: "real"   and f :: "nat \<Rightarrow> 'a" assumes "0 < r"   and "summable f" shows "\<exists>N. \<forall>n\<ge>N. norm (\<Sum>i. f (i + n)) < r"
is_tdghm_axioms: shows "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"
mgeneratorTr4: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and I :: "'a set" assumes "0 < n"   and "distinct_pds K n P"   and "ideal (O\<^bsub>K P n\<^esub>) I"   and "I \<noteq> {\<zero>\<^bsub>O\<^bsub>K P n\<^esub>\<^esub>}"   and "I \<noteq> carrier (O\<^bsub>K P n\<^esub>)" shows "mprod_exp K (mL K P I) (Kb\<^bsub>K n P\<^esub>) n \<in> carrier (O\<^bsub>K P n\<^esub>)"
ptrm_prm_apply_id: fixes X :: "'b ptrm" shows "\<epsilon> \<bullet> X = X"
verts_add_arc: fixes G :: "('c, 'd) pre_digraph"   and a :: "'d" assumes "tail G a \<in> verts G"   and "head G a \<in> verts G" shows "verts (pre_digraph.add_arc G a) = verts G"
sum_left_ideal: fixes X :: "'a set"   and U :: "'a set"   and f :: "'a \<Rightarrow> 'a" assumes "left_ideal X"   and "U \<subseteq> X"   and "finite U" shows "(\<Sum>i\<in>U. f i * i) \<in> X"
oarrivemsg_simps_6: fixes P :: "(nat \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'd"   and i' :: "nat"   and d :: "nat"   and di :: "nat" shows "oarrivemsg P \<sigma> (i':newpkt(d, di)) = True"
trss_bot_proc_const: fixes s :: "'n list"   and u :: "'n"   and c :: "'n list multiset"   and w :: "('p, 'ba) label list"   and s' :: "'n list"   and u' :: "'n"   and c' :: "'n list multiset" assumes "((s @ [u], c), w, s' @ [u'], c') \<in> trcl (trss fg)" shows "proc_of fg u = proc_of fg u'"
rupture_surj_hom_1: fixes K :: "'a set"   and p :: "'a list" assumes "subring K R"   and "p \<in> carrier (K [X])" shows "rupture_surj K p \<in> ring_hom (K [X]) (Rupt K p)"
card_prop_mono: fixes m :: "nat"   and n :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "m \<le> n" shows "card {i. i \<le> m \<and> P i} \<le> card {i. i \<le> n \<and> P i}"
prj_chine_assoc'_3: shows "fgf.Prj\<^sub>0 \<cdot> fgf.chine_assoc' = fgf.Prj\<^sub>0\<^sub>0"
llist_all2_lfilterI: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c llist"   and ys :: "'d llist"   and Q1 :: "'c \<Rightarrow> bool"   and Q2 :: "'d \<Rightarrow> bool" assumes "llist_all2 P xs ys"   and "\<And>x y. P x y \<Longrightarrow> Q1 x = Q2 y" shows "llist_all2 P (lfilter Q1 xs) (lfilter Q2 ys)"
disjointI: fixes A :: "'c set set" assumes "\<And>a b. \<lbrakk>a \<in> A; b \<in> A; a \<noteq> b\<rbrakk> \<Longrightarrow> a \<inter> b = {}" shows "disjoint A"
Un_upper1: fixes A :: "'b set"   and B :: "'b set" shows "(A::'b set) \<subseteq> A \<union> (B::'b set)"
R\<^sub>1_in_hom: shows "Maps.in_hom R\<^sub>1 (Maps.MkIde (src (tab\<^sub>0 r))) (Maps.MkIde (trg r))"
degree_pseudo_mod_less: fixes q :: "'b poly"   and p :: "'b poly" assumes "q \<noteq> 0"   and "pseudo_mod p q \<noteq> 0" shows "degree (pseudo_mod p q) < degree q"
cccFix_choose_cong: fixes heap2 :: "(Vars.var \<times> exp) list"   and cccexp1 :: "exp \<Rightarrow> Arity \<rightarrow> (Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls"   and cccexp2 :: "exp \<Rightarrow> Arity \<rightarrow> (Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls"   and heap1 :: "(Vars.var \<times> exp) list" assumes "\<And>e. e \<in> snd ` set heap2 \<Longrightarrow> cccexp1 e = cccexp2 e"   and "heap1 = heap2" shows "CoCallArityAnalysis.cccFix_choose cccexp1 heap1 = CoCallArityAnalysis.cccFix_choose cccexp2 heap2"
lower: fixes b :: "'a" assumes "b \<in> B" shows "\<exists>a\<in>A. a \<le> b"
in_mono: fixes A :: "'e set"   and B :: "'e set"   and x :: "'e" assumes "A \<subseteq> B" shows "x \<in> A \<longrightarrow> x \<in> B"
mkId1: fixes h :: "(Location \<times> Class \<times> (Field \<times> Val) list) list" shows "mkId h \<in> Pbij"
induct: fixes P :: "'a gexp \<Rightarrow> 'a gexp \<Rightarrow> bool"   and a0 :: "'a gexp"   and a1 :: "'a gexp" assumes "\<And>b1 b2. P (Bc b1) (Bc b2)"   and "\<And>b1 v va. P (Bc b1) (Eq v va)"   and "\<And>b1 v va. P (Bc b1) (Gt v va)"   and "\<And>b1 v va. P (Bc b1) (In v va)"   and "\<And>b1 v va. P (Bc b1) (Nor v va)"   and "\<And>e1 e2 b2. P (Eq e1 e2) (Bc b2)"   and "\<And>e1 e2 e1' e2'. P (Eq e1 e2) (Eq e1' e2')"   and "\<And>e1 e2 v va. P (Eq e1 e2) (Gt v va)"   and "\<And>e1 e2 v va. P (Eq e1 e2) (In v va)"   and "\<And>e1 e2 v va. P (Eq e1 e2) (Nor v va)"   and "\<And>e1 e2 b2. P (Gt e1 e2) (Bc b2)"   and "\<And>e1 e2 e1' e2'. P (Gt e1 e2) (Eq e1' e2')"   and "\<And>e1 e2 e1' e2'. P (Gt e1 e2) (Gt e1' e2')"   and "\<And>e1 e2 v va. P (Gt e1 e2) (In v va)"   and "\<And>e1 e2 v va. P (Gt e1 e2) (Nor v va)"   and "\<And>vb vc v va. P (In vb vc) (Nor v va)"   and "\<And>vb vc v va. P (In vb vc) (In v va)"   and "\<And>vb vc v. P (In vb vc) (Bc v)"   and "\<And>vb vc v va. P (In vb vc) (Eq v va)"   and "\<And>vb vc v va. P (In vb vc) (Gt v va)"   and "\<And>g1 g2 g1' g2'. \<lbrakk>P g1 g1'; P g2 g2'\<rbrakk> \<Longrightarrow> P (Nor g1 g2) (Nor g1' g2')"   and "\<And>g1 g2 v. P (Nor g1 g2) (Bc v)"   and "\<And>g1 g2 v va. P (Nor g1 g2) (Eq v va)"   and "\<And>g1 g2 v va. P (Nor g1 g2) (Gt v va)"   and "\<And>g1 g2 v va. P (Nor g1 g2) (In v va)" shows "P a0 a1"
nat_pow_zero: fixes n :: "nat" assumes "n \<noteq> 0" shows "(0::'d) ^ n = (0::'d)"
good_item_simps_67: fixes Xb :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and rhob :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option" assumes "good Xb"   and "goodEnv rhob"   and "\<And>zs z. freshEnv zs z rhob \<or> fresh zs z Xb" shows "Xb #[rhob] = Xb"
h: shows "bv_to_nat w1 \<le> 2 ^ length w1 - 1 \<and> bv_to_nat w2 \<le> 2 ^ length w2 - 1"
moebius_translation_def: fixes v :: "complex" shows "moebius_translation v = moebius_similarity 1 v"
simps_2: fixes x21 :: "'a"   and x22 :: "'a"   and y21 :: "'a"   and y22 :: "'a" shows "(Edge x21 x22 = Edge y21 y22) = (x21 = y21 \<and> x22 = y22)"
Cons_2: fixes m :: "'node"   and u' :: "('node, ('var \<times> 'node \<times> Def) set) mapping"   and p' :: "('node \<times> 'var, ('var \<times> 'node \<times> Def) list) mapping" assumes "m \<noteq> n"   and "set us \<subseteq> uses g n"   and "Mapping.lookup u n = None"   and "foldr (aux_1 g n) us (u, p) = (u', p')"   and "finite (Mapping.keys p)"   and "\<And>n v. (n, v) \<in> Mapping.keys p \<Longrightarrow> length (predecessors g n) \<noteq> 1"   and "\<And>v. Mapping.lookup p (Entry g, v) \<in> {None, Some []}" shows "Mapping.lookup u' m = Mapping.lookup u m"
fds_moebius_commute: fixes x :: "'b fds" shows "x * fds moebius_mu = fds moebius_mu * x"
is_offending_flows_def: fixes f :: "('a \<times> 'a) set"   and G :: "'a graph"   and nP :: "'a \<Rightarrow> 'b" shows "is_offending_flows f G nP \<equiv> \<not> sinvar G nP \<and> sinvar (delete_edges G f) nP"
negligible_outer_le: fixes S :: "'a set" shows "negligible S = (\<forall>e>0. \<exists>T. S \<subseteq> T \<and> T \<in> lmeasurable \<and> Sigma_Algebra.measure lebesgue T \<le> e)"
subset_diff: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and B :: "'c set"   and A :: "'c set"   and g :: "'c \<Rightarrow> 'b" assumes "abel_semigroup_set f z"   and "B \<subseteq> A"   and "finite A" shows "abel_semigroup_set.F f z g A = f (abel_semigroup_set.F f z g (A - B)) (abel_semigroup_set.F f z g B)"
hyps: shows "snd (foldl Gauss_Jordan_column_k_iarrays_PA (B, 0, A) [0..<k]) = foldl Gauss_Jordan_column_k_iarrays (0, A) [0..<k]"
fresh_3: fixes a :: "name"   and x3 :: "ty"   and x1 :: "name"   and x2 :: "trm" shows "a \<sharp> (\<lambda>x1:x3. x2) = (a \<sharp> x3 \<and> a \<sharp> [x1].x2)"
type_copy_id': shows "type_definition (\<lambda>x. x) (\<lambda>x. x) UNIV"
density_unique_iff: fixes f :: "'a \<Rightarrow> ennreal"   and f' :: "'a \<Rightarrow> ennreal" assumes "f \<in> borel_measurable (density M (\<lambda>x. 1 + h x))"   and "f' \<in> borel_measurable (density M (\<lambda>x. 1 + h x))" shows "(density (density M (\<lambda>x. 1 + h x)) f = density (density M (\<lambda>x. 1 + h x)) f') = (AE x in density M (\<lambda>x. 1 + h x). f x = f' x)"
diff: shows "zs \<noteq> ys \<or> z \<notin> {y, y1}"
nz: fixes z :: "complex" assumes "z \<in> S" shows "f z \<noteq> 0"
Required_bij_on_trancl: shows "bij_on_trancl (\<sqsubset>) Required"
step'_3: fixes c :: "int" shows "LLL_measure i (fs[i := fs ! i - c \<cdot>\<^sub>v fs ! j]) = LLL_measure i fs"
disjoint_family_on_vimageI: fixes F :: "'c \<Rightarrow> 'd set"   and I :: "'c set"   and f :: "'e \<Rightarrow> 'd" assumes "disjoint_family_on F I" shows "disjoint_family_on (\<lambda>i. f -` F i) I"
S1_mono: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "{..<m} \<times> {..<m} \<subseteq> {..<n} \<times> {..<n}"
finite_gpv_Done: fixes \<I> :: "('d, 'e) \<I>"   and x :: "'f" shows "finite_gpv \<I> (Generative_Probabilistic_Value.Done x)"
a1: shows "low_equal s1 s2 \<and> t1 = mem_mod 10 addr val s1 \<and> t2 = mem_mod 10 addr val s2"
assms: shows "M -s r \<down>m {#\<alpha>#} = {#}"
eq: shows "v \<circ> simple ct = F_sup (v \<circ> simple ct)"
length_list_update: fixes xs :: "'k list"   and i :: "nat"   and x :: "'k" shows "length (xs[i := x]) = length xs"
wlsAbs_fresh_cases: fixes xs :: "'varSort"   and s :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set"   and P :: "('index, 'bindex, 'varSort, 'var, 'opSym, 'sort) paramS"   and phi :: "'varSort \<times> 'sort \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym, 'sort) paramS \<Rightarrow> bool" assumes "wlsAbs (xs, s) A"   and "wlsPar P"   and "\<And>x X. \<lbrakk>wls s X; \<And>ys. x \<notin> varsOfS P ys; \<And>s' Y. Y \<in> termsOfS P s' \<Longrightarrow> fresh xs x Y; \<And>us s' B. B \<in> absOfS P (us, s') \<Longrightarrow> freshAbs xs x B; \<And>rho. rho \<in> envsOfS P \<Longrightarrow> freshEnv xs x rho\<rbrakk> \<Longrightarrow> phi (xs, s) (Abs xs x X) P" shows "phi (xs, s) A P"
Int_iff: fixes c :: "'c"   and A :: "'c set"   and B :: "'c set" shows "(c \<in> A \<inter> B) = (c \<in> A \<and> c \<in> B)"
fundantivertex_fundchamber_decomp: fixes s :: "'a permutation" assumes "s \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})" shows "C0 = insert (fundantivertex s) (C0 \<inter> s `\<rightarrow> C0)"
f_def: shows "\<forall>B\<in>M. f B \<in> {u \<in> U. real c / 2 < w u} \<and> f B \<in> B"
intro: fixes \<gamma> :: "'a \<Rightarrow> int set"   and num' :: "int \<Rightarrow> 'a"   and plus' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "Val_abs \<gamma> num' plus'" shows "Gamma num' plus' \<gamma>"
bicategory_of_spans_respects_equivalence: fixes V\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>C :: "'c \<Rightarrow> 'c"   and src\<^sub>C :: "'c \<Rightarrow> 'c"   and trg\<^sub>C :: "'c \<Rightarrow> 'c"   and V\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>D :: "'d \<Rightarrow> 'd"   and src\<^sub>D :: "'d \<Rightarrow> 'd"   and trg\<^sub>D :: "'d \<Rightarrow> 'd" assumes "equivalent_bicategories 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"   and "bicategory_of_spans V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C" shows "bicategory_of_spans V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D"
bij_betw_nth: fixes xs :: "'c list"   and A :: "nat set"   and B :: "'c set" assumes "cycle xs"   and "A = {..<length xs}"   and "B = set xs" shows "bij_betw ((!) xs) A B"
fix_ind: fixes P :: "'d \<Rightarrow> bool"   and F :: "'d \<rightarrow> 'd" assumes "adm P"   and "P \<bottom>"   and "\<And>x. P x \<Longrightarrow> P (F\<cdot>x)" shows "P (fix\<cdot>F)"
digraph_isomorphism_invI: fixes hom :: "('a, 'a \<times> nat \<times> 'a, 'c, 'd) digraph_isomorphism" assumes "H.digraph_isomorphism hom" shows "pre_digraph.digraph_isomorphism (app_iso hom (app_iso iso_f G)) (H.inv_iso hom)"
vcons_vsingleton: fixes a :: "V" shows "[a]\<^sub>\<circ> = set {\<langle>0\<^sub>\<nat>, a\<rangle>}"
assms_4: shows "(r, Store e2 mo r) \<in> set (node_deliver_messages xs)"
connected_distI: fixes v :: "nat"   and d :: "nat"   and v' :: "nat" assumes "cf.dist v d v'" shows "cf.connected v v'"
is_generalisation_of_preserves_reg: fixes t' :: "transition"   and t :: "transition"   and i :: "nat"   and r :: "nat"   and ia :: "value list"   and c :: "nat \<Rightarrow>f value option" assumes "is_generalisation_of t' t i r" shows "evaluate_updates t ia c $ r = c $ r"
enum_S: fixes x :: "'a" assumes "x \<in> S" shows "\<exists>n. x \<in> F (a n) \<and> x \<in> G (a n)"
exec_step_ind_cases_3: fixes v :: "val"   and P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap"   and \<sigma> :: "jvm_state" assumes "exec_step_ind (StepI (Push v)) P h stk loc C M pc ics frs sh \<sigma>" obtains "\<sigma> = (None, h, (v # stk, loc, C, M, Suc pc, ics) # frs, sh)"
eq_refl: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k"   and y :: "'k" assumes "class.preorder less_eq less"   and "x = y" shows "less_eq x y"
IN_plus_current: fixes x :: "'v" shows "d_IN (plus_current f g) x = (if x \<in> RF (TER f) then d_IN f x else d_IN g x)"
chine_composite_2: shows "\<mu>\<nu>\<pi>.chine = \<mu>.leg0 \<down>\<down> \<nu>.leg1 \<cdot> \<nu>\<pi>.prj\<^sub>1"
rPar1: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>Q :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list"   and C :: "'d" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<And>C. Prop C (\<Psi> \<otimes> \<Psi>\<^sub>Q) P M (\<lparr>\<nu>*xvec\<rparr>N \<prec>' P')"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* M"   and "A\<^sub>Q \<sharp>* xvec"   and "A\<^sub>Q \<sharp>* N"   and "A\<^sub>Q \<sharp>* P'"   and "A\<^sub>Q \<sharp>* C"   and "xvec \<sharp>* Q"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* \<Psi>\<^sub>Q"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* C" shows "Prop C \<Psi> (P \<parallel> Q) M (\<lparr>\<nu>*xvec\<rparr>N \<prec>' (P' \<parallel> Q))"
Meet_lower: fixes a :: "'b"   and A :: "'b set" assumes "a \<in> A" shows "\<Sqinter>A \<sqsubseteq> a"
lower_adjoint_def: fixes A :: "('d, 'e) gorder_scheme"   and B :: "('f, 'g) gorder_scheme"   and f :: "'d \<Rightarrow> 'f" shows "lower_adjoint A B f \<equiv> \<exists>g. galois_connection \<lparr>orderA = A, orderB = B, lower = f, upper = g\<rparr>"
pivotandupdate_valuation_xi: fixes eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and pivot_and_update :: "nat \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> ('b, 'a) state \<Rightarrow> ('b, 'a) state"   and s :: "('b, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and c :: "'a" assumes "PivotAndUpdate eq_idx_for_lvar pivot_and_update"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (EqForLVar.eq_for_lvar eq_idx_for_lvar (\<T> s) x\<^sub>i)" shows "Mapping.lookup (\<V> (pivot_and_update x\<^sub>i x\<^sub>j c s)) x\<^sub>i = Some c"
fps_inverse_eq_0_iff': fixes f :: "'a fps" shows "(inverse f = 0) = (inverse (fps_nth f 0) = (0::'a))"
assms: shows "\<exists>x. \<^bold>\<not> (E x)"
create_closure: fixes x :: "'b \<Longrightarrow> 'c"   and v :: "'b" assumes "weak_lens x" shows "create\<^bsub>x\<^esub> v \<in> \<S>\<^bsub>x\<^esub>"
alphaBinp_sym: fixes qbinp :: "'d \<Rightarrow> ('e, 'd, 'varSort, 'var, 'f) qAbs option"   and qbinp' :: "'d \<Rightarrow> ('e, 'd, 'varSort, 'var, 'f) qAbs option" assumes "(qbinp, qbinp') \<in> alphaBinp" shows "(qbinp', qbinp) \<in> alphaBinp"
FAssNull\<^sub>1: fixes P :: "nat exp prog"   and e\<^sub>1 :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap"   and e\<^sub>2 :: "nat exp"   and v :: "val"   and s\<^sub>2 :: "heap \<times> val list \<times> sheap"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>2\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1\<bullet>F{D} := e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW NullPointer,s\<^sub>2\<rangle>"
prefix_prefix: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "prefix xs ys" shows "prefix xs (ys @ zs)"
foldr_Nil: fixes f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd" shows "foldr f [] = id"
srev_induct: fixes P :: "[:'a:] \<Rightarrow> bool"   and xs :: "[:'a:]" assumes "P \<bottom>"   and "P [::]"   and "\<And>x xs. \<lbrakk>x \<noteq> \<bottom>; xs \<noteq> \<bottom>; P xs\<rbrakk> \<Longrightarrow> P (xs :@ [:x:])" shows "P xs"
same_bound: fixes C :: "nat set"   and f :: "nat \<Rightarrow> real"   and g :: "nat \<Rightarrow> real" assumes "C \<subseteq> PR"   and "np \<le> card C + khl"   and "3 * khl < np" shows "\<exists>i\<in>C. Min (reduce f PR) \<le> f i \<and> g i \<le> Max (reduce g PR)"