hcju/isabelleps huggingface dataset:
cut_less_Int_right: fixes A :: "'a set"   and B :: "'a set"   and t :: "'a" shows "(A \<inter> B) \<down>< t = A \<inter> B \<down>< t"
sort_remove1_eq: fixes cmp :: "'b comparator"   and x :: "'b"   and xs :: "'b list" shows "Sorting_Algorithms.sort cmp (remove1 x xs) = remove1 x (Sorting_Algorithms.sort cmp xs)"
dg_Par_is_arrD_3: fixes \<alpha> :: "V"   and A :: "V"   and B :: "V"   and S :: "V" assumes "S : A \<mapsto>\<^bsub>dg_Par \<alpha>\<^esub> B" shows "S\<lparr>ArrCod\<rparr> = B"
no_calls2_def: fixes es :: "'a expr1 list"   and pc :: "nat" shows "no_calls2 es pc = (pc \<le> length (compEs2 es) \<and> (pc < length (compEs2 es) \<longrightarrow> (\<forall>M n. compEs2 es ! pc \<noteq> Invoke M n)))"
integrable_zero: fixes M :: "'c measure" shows "integrable M (\<lambda>x. 0::'d)"
P_of_state_Liminf: shows "P_of_state (Liminf_state Sts) = Liminf_llist (lmap P_of_state Sts)"
listrelp_betas: fixes ts :: "dB list"   and ts' :: "dB list"   and t :: "dB"   and t' :: "dB" assumes "listrelp (\<rightarrow>\<^sub>\<beta>\<^sup>*) ts ts'"   and "t \<rightarrow>\<^sub>\<beta>\<^sup>* t'" shows "t \<degree>\<degree> ts \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<degree>\<degree> ts'"
ts_inv_ok_upd_inv: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and I :: "'f \<Rightarrow> 'i option"   and ta :: "('f, 'g, 'j) new_thread_action"   and P :: "'i \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> 'j \<Rightarrow> bool" assumes "ts_inv_ok ts I" shows "ts_inv_ok (redT_updT ts ta) (upd_inv I P ta)"
linear': fixes a :: "'a"   and b :: "'a" assumes "a \<in> S"   and "b \<in> S"   and "a \<noteq> b" shows "le a b \<or> le b a"
compxEs2_size_convs: fixes n :: "nat"   and es :: "'a expr1 list"   and d :: "nat" assumes "n \<noteq> 0" shows "compxEs2 es n d = shift n (compxEs2 es 0 d)"
abs_eq: fixes x :: "1 word \<times> 'a word \<times> 'b word" shows "fraction (Abs_float x) = (case x of (uu_, uua_, xa) \<Rightarrow> unat xa)"
listrel_app: fixes xs1 :: "'b list"   and ys1 :: "'c list"   and r :: "('b \<times> 'c) set"   and xs2 :: "'b list"   and ys2 :: "'c list" assumes "(xs1, ys1) \<in> listrel r"   and "(xs2, ys2) \<in> listrel r" shows "(xs1 @ xs2, ys1 @ ys2) \<in> listrel r"
f': assumes "x \<in> S \<union> closure S \<inter> closure T" shows "(f has_vector_derivative f' x) (at x within S \<union> closure S \<inter> closure T)"
add_same_coeff_l: fixes x :: "char list multiset"   and n :: "int"   and p :: "(char list multiset \<times> int) multiset"   and q :: "(char list multiset \<times> int) multiset"   and m :: "int"   and r :: "(char list multiset \<times> int) multiset" shows "add_poly_p (add_mset (x, n) p, q, add_mset (x, m) r) (p, q, add_mset (x, n + m) r)"
sum_upto_dcharacter_le: fixes x :: "real" assumes "\<chi> \<noteq> \<chi>\<^sub>0" shows "cmod (sum_upto \<chi> x) \<le> real (totient n)"
vector_space_on_axioms: shows "vector_space_on U\<^sub>V\<^sub>S (*s)"
ring_hom_memE_1: fixes h :: "'c \<Rightarrow> 'd"   and R :: "('c, 'e) ring_scheme"   and S :: "('d, 'f) ring_scheme"   and x :: "'c" assumes "h \<in> ring_hom R S"   and "x \<in> carrier R" shows "h x \<in> carrier S"
HStartBallot_HInv4a: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HStartBallot s s' p"   and "HInv4a s q"   and "Inv2a s'" shows "HInv4a s' q"
bconf_Try: fixes P :: "'a prog"   and sh :: "sheap"   and e\<^sub>1 :: "'b exp"   and C :: "char list"   and V :: "'b"   and e\<^sub>2 :: "'b exp"   and b :: "bool" shows "(P,sh \<turnstile>\<^sub>b (try e\<^sub>1 catch(C V) e\<^sub>2,b) \<surd>) = (P,sh \<turnstile>\<^sub>b (e\<^sub>1,b) \<surd>)"
intro: fixes get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and type_wf :: "(_) heap \<Rightarrow> bool" assumes "\<And>document_ptr h h'. reads (get_disconnected_nodes_locs document_ptr) (get_disconnected_nodes document_ptr) h h'"   and "\<And>h document_ptr. \<lbrakk>type_wf h; document_ptr |\<in>| document_ptr_kinds h\<rbrakk> \<Longrightarrow> h \<turnstile> ok get_disconnected_nodes document_ptr"   and "\<And>h document_ptr. h \<turnstile> ok get_disconnected_nodes document_ptr \<Longrightarrow> document_ptr |\<in>| document_ptr_kinds h"   and "\<And>document_ptr h. pure (get_disconnected_nodes document_ptr) h" shows "l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs"
two_neq_one_aux: fixes A :: "'a"   and B :: "'a"   and C :: "'a" assumes "(\<LM> A  \<RM>) \<oplus> B = \<LM> C  \<RM>" shows "False"
elims_3: fixes x :: "'a \<Rightarrow> bool"   and xa :: "('a, 'b) result" assumes "\<not> if_rval x xa" obtains f :: "'a \<Rightarrow> bool" and v0 :: "'a" where "x = f" and "xa = Rval v0" and "\<not> f v0"
Disj_commute_Imp: fixes H :: "fm set"   and B :: "fm"   and A :: "fm" shows "H \<turnstile> B OR A IMP A OR B"
list_order_monotonic: fixes A :: "('c \<times> 'c option) list"   and B :: "('c \<times> 'c option) list"   and x :: "'c"   and y :: "'c" assumes "insert_ops A"   and "insert_ops B"   and "set A \<subseteq> set B"   and "Insert_Spec.list_order A x y" shows "Insert_Spec.list_order B x y"
nat_of_num_mult: fixes x :: "num"   and y :: "num" shows "nat_of_num (x * y) = nat_of_num x * nat_of_num y"
dither_loops: shows "{\<lambda>tp. \<exists>k. tp = (Bk \<up> k, <0>)} dither \<up>"
per_union_related: fixes R :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a" assumes "part_equiv R"   and "a \<in> Domain R"   and "b \<in> Domain R" shows "(a, b) \<in> per_union R a b"
arrI: fixes f :: "('o, 'a) arr"   and a :: "('o, 'a) arr"   and b :: "('o, 'a) arr" assumes "in_hom f a b" shows "arr f"
length_of_bin_rep: fixes m :: "nat"   and n :: "nat" assumes "m < 2 ^ n" shows "length (bin_rep n m) = n"
is_sig_red_cong: fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set" assumes "lt p = lt q"   and "rep_list p = rep_list q" shows "is_sig_red sing_reg top_tail F p = is_sig_red sing_reg top_tail F q"
ssubst_insert_if: fixes V :: "name set"   and t :: "tm"   and i :: "name" assumes "finite V" shows "ssubst t (insert i V) F = (if i \<in> V then ssubst t V F else subst i (F i) (ssubst t V F))"
negligible_insert: fixes a :: "'b"   and s :: "'b set" shows "negligible (insert a s) = negligible s"
list_wset_rel_def: fixes R :: "('a \<times> 'b) set" shows "\<langle>R\<rangle>list_wset_rel = br set top O \<langle>R\<rangle>set_rel"
arrow_of_spans_axioms: shows "arrow_of_spans (\<cdot>) \<mu>'"
scale_zero_left: fixes x :: "'c" shows "(0::'c) * x = (0::'c)"
scaleRat_uminus: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleRat :: "rat \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.linordered_rational_vector less_eq less plus zero minus uminus scaleRat" shows "scaleRat (- 1) x = uminus x"
compl_eq_compl_iff: fixes x :: "'a"   and y :: "'a" shows "(- x = - y) = (x = y)"
sr: shows "s \<in> reachable (closed (pnet np n)) TT"
that_2: fixes st :: "'St"   and u :: "'U"   and c :: "'C"   and sl1 :: "('St, 'U, 'C) state list" assumes "sl = State st u c # sl1"   and "sl1 \<noteq> [] \<Longrightarrow> wffp sl1 \<and> hd sl1 \<in> \<delta> (State st u c)" shows "thesis"
correctCompositionKS_exprChannel_s_Pex: fixes PQ :: "specID"   and P :: "specID"   and Q :: "specID"   and secret :: "Secrets"   and ch :: "chanID" assumes "subcomponents PQ = {P, Q}"   and "correctCompositionKS PQ"   and "sKS secret \<notin> LocalSecrets PQ"   and "ch \<in> ins P"   and "exprChannel ch (sE secret)"   and "sKS secret \<notin> specKeysSecrets PQ"   and "correctCompositionIn PQ" shows "\<exists>ch. ch \<in> ins PQ \<and> exprChannel ch (sE secret)"
assms: shows "\<forall>v. \<exists>v'. (\<forall>x\<in>vars p. v' x = v x) \<and> p \<lbrace> v' \<rbrace> = (0::'a)"
union: fixes x :: "ZF"   and A :: "ZF"   and B :: "ZF" shows "Elem x (HOLZF.union A B) = (Elem x A \<or> Elem x B)"
HBL1_def: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and P :: "'c" shows "HBL1 var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv enc P \<equiv> Encode var trm fmla Var FvarsT substT Fvars subst num enc \<and> Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv \<and> HBL1_axioms var fmla Fvars subst prv enc P"
const_times_polyn: fixes c :: "'a"   and p :: "'a" assumes "Idomain S"   and "c \<in> carrier S"   and "c \<noteq> \<zero>\<^bsub>S\<^esub>"   and "p \<in> carrier R"   and "p \<noteq> \<zero>" shows "c \<cdot>\<^sub>r p \<noteq> \<zero> \<and> deg_n R S X (c \<cdot>\<^sub>r p) = deg_n R S X p"
bool_lists_card: fixes m :: "nat" shows "real (card {bl. length bl = m}) = b ^ m"
span_insert_zero: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "pmdl (insert 0 B) = pmdl B"
mask_out_sub_mask: fixes x :: "'a word"   and n :: "nat" shows "x && ~~ mask n = x - (x && mask n)"
two_dvd_exp: fixes x :: "nat" assumes "0 < x" shows "even (2 ^ x)"
pi: fixes e :: "real" assumes "0 < e" shows "\<exists>p. valid_path p \<and> z \<notin> path_image p \<and> pathfinish p = pathstart p \<and> (\<forall>t\<in>{0..1}. cmod (\<gamma> t - p t) < e) \<and> contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * n"
X_def: shows "X = (\<lambda>i f. if f i then 1 else 0)"
X_1: fixes e :: "'b" assumes "e \<in> arcs G" shows "tail H e \<in> verts G"
contractible_imp_path_connected: fixes S :: "'b set" assumes "contractible S" shows "path_connected S"
normal_series_subgroups: fixes G :: "('c, 'd) monoid_scheme"   and \<GG> :: "'c set list"   and i :: "nat" assumes "normal_series G \<GG>"   and "i < length \<GG>" shows "subgroup (\<GG> ! i) G"
count_0: fixes B :: "'b list" shows "(\<forall>x. PermutationLemmas.count x B = 0) = (B = [])"
domainI': fixes x :: "'b"   and y :: "'b" assumes "x \<sim> y" shows "x \<in> domain"
intros_2: fixes x :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)"   and y :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)"   and n :: "nat"   and z :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)" assumes "x \<leadsto>\<^sub>A y"   and "abs.neval y n z" shows "abs.neval x (Suc n) z"
mat_of_row_carrier_2: fixes y :: "'b vec"   and n :: "nat" assumes "y \<in> carrier_vec n" shows "mat_of_row y \<in> carrier_mat (Suc 0) n"
full_def: fixes \<sigma> :: "'tp" shows "Ik.full \<sigma> \<equiv> \<forall>d. Ik_intT \<sigma> d"
inner_commute_zero: fixes xa :: "'b"   and x :: "'b" shows "(inner xa x = (0::'a)) = (inner x xa = (0::'a))"
valid_weaken_pre: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and P' :: "'e set" assumes "\<Gamma> \<Turnstile>\<^bsub>/F\<^esub> P c Q, A"   and "P' \<subseteq> P" shows "\<Gamma> \<Turnstile>\<^bsub>/F\<^esub> P' c Q, A"
consistent_eq2: fixes \<sigma> :: "'c \<Rightarrow> 'd"   and \<sigma>\<^sub>s\<^sub>y\<^sub>m :: "'c \<times> nat \<Rightarrow> 'd"   and store :: "'c \<Rightarrow> nat" shows "consistent \<sigma> \<sigma>\<^sub>s\<^sub>y\<^sub>m store = (\<sigma> = (\<lambda>v. \<sigma>\<^sub>s\<^sub>y\<^sub>m (symvar v store)))"
axioms_1: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and b :: "'ah" assumes "constant_functor A B b" shows "category A"
lists_image: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "lists (f ` A) = map f ` lists A"
star_inductl': fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z + x \<cdot> y \<le> y" shows "x\<^sup>\<star> \<cdot> z \<le> y"
i_join_i_drop: fixes I :: "nat set"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "I \<noteq> {}" shows "f \<Up> n \<Join>\<^sub> I = f \<Join>\<^sub> (I \<oplus> n)"
monom_mult_code: fixes n :: "nat"   and f :: "'b poly" shows "coeffs (monom_mult n f) = (let xs = coeffs f in if xs = [] then xs else replicate n (0::'b) @ xs)"
w_in_hom_1: shows "\<guillemotleft>w : src r\<^sub>0 \<rightarrow> src s\<^sub>0\<guillemotright>"
le_left_comm: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Le Bet Cong A B C D" shows "Tarski_neutral_dimensionless.Le Bet Cong B A C D"
dyn_weak_control_dependence_def: fixes n :: "'node"   and n' :: "'node"   and as :: "'edge list" shows "n weakly controls n' via as \<equiv> \<exists>a a' as'. as = a # as' \<and> n' \<notin> set (sourcenodes as) \<and> n -as\<rightarrow>* n' \<and> n' strongly-postdominates targetnode a \<and> valid_edge a' \<and> sourcenode a' = n \<and> \<not> n' strongly-postdominates targetnode a'"
f_def: fixes i :: "nat" shows "local.f i = fs ! i"
in_sub_in2: shows "inside (path_image c2 \<union> path_image c) \<subseteq> inside (path_image c1 \<union> path_image c2)"
project_enc_extend: fixes w :: "'a list"   and I :: "(nat + nat set) list"   and x :: "('a \<times> bool list) list"   and m :: "nat" assumes "wf_interp w I"   and "fin_cut_same (any, replicate (length I) False) (map \<pi> x) @ replicate m (any, replicate (length I) False) \<in> enc (w, I)"   and "\<forall>(uu_, x)\<in>set x. x \<noteq> []" shows "x \<in> enc (w, Inr (positions_in_row (x @- sconst (any, replicate (Suc (length I)) False)) 0) # I)"
axioms_2: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set" assumes "calculus Bot Inf entails Red_I Red_F" shows "calculus_axioms Bot Inf entails Red_I Red_F"
h2: shows "take_rows_from_matrix (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets))) (rows_to_keep (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))) = take_rows_from_matrix (take_cols_from_matrix (M_mat_R signs subsets) (find_nonzeros_from_input_vec (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))) (rows_to_keep (take_cols_from_matrix (M_mat_R signs subsets) (find_nonzeros_from_input_vec (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))))"
induct: fixes P :: "('d \<times> ('d, 'c) RHS) list \<Rightarrow> 'c \<Rightarrow> bool"   and a0 :: "('d \<times> ('d, 'c) RHS) list"   and a1 :: "'c" assumes "\<And>t. P [] t"   and "\<And>X A B ps t. P ps t \<Longrightarrow> P ((X, Branch A B) # ps) t"   and "\<And>X a ps t. \<lbrakk>a = t \<Longrightarrow> P ps t; a \<noteq> t \<Longrightarrow> P ps t\<rbrakk> \<Longrightarrow> P ((X, Leaf a) # ps) t" shows "P a0 a1"
laplace_expansion_row: fixes A :: "'a mat"   and n :: "nat"   and i :: "nat" assumes "A \<in> carrier_mat n n"   and "i < n" shows "Determinant.det A = (\<Sum>j<n. A $$ (i, j) * Determinant.cofactor A i j)"
unit_factor_eq_0_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "(unit_factor a = zero) = (a = zero)"
simps: fixes A :: "('b, unit) Old_Datatype.node set \<Rightarrow> bool"   and a :: "('b, unit) Old_Datatype.node set" shows "listp A a = (a = NIL \<or> (\<exists>aa M. a = CONS aa M \<and> A aa \<and> listp A M))"
dominates_def: fixes g :: "'g"   and n :: "'node"   and m :: "'node" shows "dominates g n m \<equiv> m \<in> set (\<alpha>n g) \<and> (\<forall>ns. g \<turnstile> Entry g-ns\<rightarrow>m \<longrightarrow> n \<in> set ns)"
deutsch_algo_eval_def: shows "deutsch_algo_eval \<equiv> prob1 2 deutsch_algo 0"
map_val_raw_cong: fixes xs :: "('e \<times> 'f) list \<times> 'g"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'h"   and g :: "'e \<Rightarrow> 'f \<Rightarrow> 'h" assumes "\<And>k v. (k, v) \<in> set (fst xs) \<Longrightarrow> f k v = g k v" shows "map_val_raw f xs = map_val_raw g xs"
abc_steps_add: fixes as :: "nat"   and lm :: "nat list"   and ap :: "abc_inst list"   and m :: "nat"   and n :: "nat" shows "abc_steps_l (as, lm) ap (m + n) = abc_steps_l (abc_steps_l (as, lm) ap m) ap n"
contI: fixes f :: "'b \<Rightarrow> 'c" assumes "\<And>Y. chain Y \<Longrightarrow> range (\<lambda>i. f (Y i)) <<| f (\<Squnion>i. Y i)" shows "cont f"
pseudo_proj_False_preimage': fixes g :: "bool stream \<Rightarrow> 'a"   and n :: "nat"   and N :: "'a measure"   and z :: "bool stream" assumes "g \<in> F n \<rightarrow>\<^sub>M N"   and "set_discriminating n g N" shows "pseudo_proj_False n -` g -` {g z} = g -` {g z}"
assms_2: shows "\<exists>f. \<forall>i. (f i, f (Suc i)) \<in> S"
strong_induct: fixes P :: "'a \<Rightarrow> ty \<Rightarrow> bool"   and z :: "'a"   and ty :: "ty" assumes "\<And>z. P z TBase"   and "\<And>ty1 ty2 z. \<lbrakk>\<And>z. P z ty1; \<And>z. P z ty2\<rbrakk> \<Longrightarrow> P z (ty1 \<rightarrow> ty2)"   and "\<And>ty z. (\<And>z. P z ty) \<Longrightarrow> P z (T ty)" shows "P z ty"
hyps: fixes A :: "(('a, 'n) vec, 'n) vec"   and P :: "(('a, 'n) vec, 'n) vec" assumes "n < nrows A" shows "det (fst (row_add_iterate_PA (P, A) n i j)) = det P"
nres_order_simps_8: fixes X :: "'b nres set" shows "(FAIL = Sup X) = (FAIL \<in> X)"
elims_2: fixes x :: "bool list"   and xa :: "tree" assumes "internal x xa" obtains l :: "tree" and r :: "tree" where "x = []" and "xa = Branching l__ r__"   | d :: "bool" and ds :: "bool list" and l :: "tree" and r :: "tree" where "x = d # ds" and "xa = Branching l r" and "if d then internal ds l else internal ds r"
cont_tf_thm_1: fixes v :: "i" shows "[ContingentlyTrue q\<^sub>0 \<^bold>\<or> ContingentlyFalse q\<^sub>0 in v]"
arbitrary_union_of_idempot: fixes P :: "'c set \<Rightarrow> bool" shows "arbitrary union_of arbitrary union_of P = arbitrary union_of P"
zero_im_sheaf_def: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and S' :: "'e set"   and is_open' :: "'e set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e" assumes "Comm_Ring.im_sheaf S is_open \<FF> \<rho> b add_str mult_str zero_str one_str S' is_open' f" shows "im_sheaf.zero_im_sheaf S zero_str f \<equiv> \<lambda>V. zero_str (f \<^sup>\<inverse> S V)"
th2_2: fixes a :: "hyp2"   and b :: "hyp2"   and c :: "hyp2"   and d :: "hyp2" assumes "a b \<congruent>\<^sub>K c d" shows "c d \<congruent>\<^sub>K a b"
UNlift01_shift: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('test, 'atom, 'choice) cmd list"   and P :: "nat \<Rightarrow> nat set set"   and n :: "nat"   and I :: "nat set"   and F :: "nat \<Rightarrow> nat set \<Rightarrow> nat set" assumes "length cl = length dl"   and "\<And>n. n < length cl \<Longrightarrow> part {..<brn (cl ! n)} (P n) \<and> {} \<notin> P n"   and "n \<in> theNFTBoth cl dl"   and "I \<in> P n" shows "UNlift01 cl dl P F (local.shift cl n I) = local.shift dl n (F n I)"
RETURN_refine_iff: fixes x :: "'a"   and R :: "('a \<times> 'b) set"   and y :: "'b" shows "(RETURN x \<le> \<Down> R (RETURN y)) = ((x, y) \<in> R)"
with_def: fixes zero :: "'d"   and zeroa :: "'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e" shows "dependent.with zero zeroa plus \<equiv> \<lambda>scale s. \<exists>t u. finite t \<and> t \<subseteq> s \<and> sum_with plus zeroa (\<lambda>v. scale (u v) v) t = zeroa \<and> (\<exists>v\<in>t. u v \<noteq> zero)"
complex_cone_altdef: fixes a :: "real"   and b :: "real" assumes "- pi < a"   and "a \<le> b"   and "b \<le> pi" shows "complex_cone a b = insert 0 {z. arg z \<in> {a..b}}"
HFun_Sigma_sf: fixes t :: "tm" shows "Sigma_fm (HFun_Sigma t)"
sfp: shows "F, F, \<Gamma> \<Rightarrow>\<^sub>n"
intv_CL_glb: fixes a :: "'a"   and b :: "'a" assumes "a \<in> A"   and "b \<in> A"   and "interval r a b \<noteq> {}" shows "\<forall>S\<subseteq>interval r a b. \<exists>G. isGlb S \<lparr>pset = interval r a b, order = induced (interval r a b) r\<rparr> G"
in_minimize_wrt_False: fixes Q :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and x :: "'a"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a" assumes "\<And>x y. Q x y \<Longrightarrow> \<not> Q y x"   and "sorted_wrt Q xs"   and "x \<in> set (minimize_wrt P xs)"   and "\<not> P y x"   and "Q y x"   and "y \<in> set xs"   and "y \<noteq> x" shows "False"
sort_key_simps_2: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and xs :: "'b list" shows "sort_key f (x # xs) = insort_key f x (sort_key f xs)"
setops_subterms_cases: fixes t :: "('fun, 'a) Term.term"   and S :: "('fun, 'a) stateful_strand_step list" assumes "t \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t pair ` setops\<^sub>s\<^sub>s\<^sub>t S" shows "t \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t trms\<^sub>s\<^sub>s\<^sub>t S \<or> t \<in> pair ` setops\<^sub>s\<^sub>s\<^sub>t S"
eFresh_nchotomy: fixes eX :: "'c withERR"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and ys :: "'varSort"   and y :: "'var" shows "(\<exists>s X. eX = OK X \<and> igWls MOD s X) \<or> eFresh MOD ys y eX"
assms_2: shows "A\<^sub>G \<sharp>* A\<^sub>F"
list_pad_out_def: fixes xs :: "'a list"   and k :: "nat" shows "list_pad_out xs k = xs @ replicate (k + 1 - length xs) undefined"
mor_Abs_JF: shows "mor car_final1 car_final2 str_final1 str_final2 UNIV UNIV dtor1 dtor2 Abs_JF1 Abs_JF2"
mset_size1elem: fixes P :: "'a multiset"   and q :: "'a" assumes "size P \<le> 1"   and "q \<in># P" shows "P = {#q#}"
nn_cond_exp_intg: fixes M :: "'b measure"   and F :: "'b measure"   and f :: "'b \<Rightarrow> ennreal"   and g :: "'b \<Rightarrow> ennreal" assumes "sigma_finite_subalgebra M F"   and "f \<in> borel_measurable F"   and "g \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. f x * nn_cond_exp M F g x \<partial>M = \<integral>\<^sup>+ x. f x * g x \<partial>M"
elims_3: fixes x :: "nat"   and xa :: "nat"   and xb :: "cell list \<times> cell list" assumes "\<not> wadjust_on_left_moving x xa xb" obtains m :: "nat" and rs :: "nat" and l :: "cell list" and r :: "cell list" where "x = m" and "xa = rs" and "xb = (l, r)" and "\<not> (wadjust_on_left_moving_O m rs (l, r) \<or> wadjust_on_left_moving_B m rs (l, r))"
degree_mod_less: fixes y :: "'a poly"   and x :: "'a poly" assumes "y \<noteq> 0" shows "x mod y = 0 \<or> Polynomial.degree (x mod y) < Polynomial.degree y"
vpeq_refl: fixes u :: "partition_id_t"   and s :: "state_t" shows "vpeq u s s"