hcju/isabelleps huggingface dataset:
stoch_proc_subalg_nat_filt: fixes p :: "real"   and M :: "bool stream measure"   and X :: "nat \<Rightarrow> bool stream \<Rightarrow> 'b"   and n :: "nat" assumes "infinite_coin_toss_space p M"   and "borel_adapt_stoch_proc (infinite_coin_toss_space.nat_filtration M) X" shows "subalgebra (infinite_coin_toss_space.nat_filtration M n) (stoch_proc_filt M X borel n)"
square_eq_1_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.ring_1_no_zero_divisors one times plus zero minus uminus" shows "(times x x = one) = (x = one \<or> x = uminus one)"
dvd_def: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and a :: "'d" shows "dvd.dvd times b a = (\<exists>k. a = times b k)"
mahler_measure_constant: fixes c :: "complex" shows "mahler_measure_poly [:c:] = cmod c"
prefix_length_le: fixes xs :: "'a list"   and ys :: "'a list" assumes "prefix xs ys" shows "length xs \<le> length ys"
simps_1: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d" shows "f_Exec_Comp trans_fun [] c = c"
fps_eq_iff: fixes p :: "'a fps"   and q :: "'a fps" shows "(p = q) = (\<forall>n. fps_nth p n = fps_nth q n)"
strict_trans1: fixes b :: "'b list"   and a :: "'b list"   and c :: "'b list" assumes "b \<le>p a"   and "c <p b" shows "c <p a"
noAbrupt_startn: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and n :: "nat"   and t :: "('b, 'c) xstate"   and s' :: "'b" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"   and "\<forall>t'. t \<noteq> Abrupt t'" shows "s \<noteq> Abrupt s'"
abs_eq: fixes xa :: "'d \<Rightarrow> 'e option"   and x :: "'d \<Rightarrow> 'e option" shows "mapping_add (mapping.Mapping xa) (mapping.Mapping x) = mapping.Mapping (xa ++ x)"
neq_LNil_conv: fixes xs :: "'a llist" shows "(xs \<noteq> LNil) = (\<exists>x xs'. xs = LCons x xs')"
fw_int_aux_Suc_b: fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and a :: "nat"   and b :: "nat"   and c :: "nat"   and i :: "nat"   and j :: "nat" assumes "dbm_int (fw M n a b n) n"   and "a \<le> n"   and "Suc b \<le> n"   and "c \<le> n"   and "i \<le> n"   and "j \<le> n"   and "fw M n a (Suc b) c i j \<noteq> \<infinity>" shows "get_const (fw M n a (Suc b) c i j) \<in> \<int>"
distinct_names: shows "all_distinct (Node (Node (Node Tip A False Tip) F False (Node Tip a False Tip)) b False (Node (Node Tip c False Tip) f False (Node Tip x False Tip)))"
emsapss_decode_help10_def: fixes M' :: "bit list"   and H :: "bit list" shows "emsapss_decode_help10 M' H = emsapss_decode_help11 (sha1 M') H"
preserves_guarantee_compliance: shows "preserves_local_guarantee_compliance \<R>"
assms: assumes "x \<notin>\<^sub>\<circ> B" shows "x \<in>\<^sub>\<circ> A"
prime_gauss_int_norm_imp_prime_elem: fixes q :: "gauss_int" assumes "prime (gauss_int_norm q)" shows "prime_elem q"
error_msg_ne_SUCCES_3: fixes i :: "'a"   and m :: "char list" shows "is_cfailed (error_msg i m)"
atLeastLessThan_less_eq2: fixes f :: "nat \<Rightarrow> nat"   and m :: "nat"   and n :: "nat" assumes "inj_on f {0..<m} \<and> f ` {0..<m} \<subseteq> {0..<n}" shows "m \<le> n"
him_eq: fixes a :: "octo"   and b :: "octo" shows "(HIm a = HIm b) = (Im1 a = Im1 b \<and> Im2 a = Im2 b \<and> Im3 a = Im3 b \<and> Im4 a = Im4 b \<and> Im5 a = Im5 b \<and> Im6 a = Im6 b \<and> Im7 a = Im7 b)"
a1: shows "low_equal s1 s2 \<and> load_word_mem s1 addra 10 = load_word_mem s2 addra 10"
gr0_implies_Suc: fixes n :: "nat" assumes "order.greater n 0" shows "\<exists>m. n = Suc m"
lmap_conv_unfold_llist: fixes f :: "'a \<Rightarrow> 'b" shows "lmap f = unfold_llist (\<lambda>xs. xs = LNil) (f \<circ> lhd) ltl"
assms_1: shows "m \<triangleleft>\<triangleright>\<^sup>* m'"
trivial_homology_group_empty: fixes X :: "'b topology"   and p :: "int" assumes "topspace X = {}" shows "trivial_group (homology_group p X)"
distinct_7: fixes x21 :: "'d rexp"   and x22 :: "'d rexp"   and x31 :: "'d rexp"   and x32 :: "'d rexp" shows "Alt x21 x22 \<noteq> Conc x31 x32"
tset_ttl: fixes xs :: "('a, 'b) tllist" shows "tset (ttl xs) \<subseteq> tset xs"
add_neg_neg: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less a zero"   and "less b zero" shows "less (plus a b) zero"
Rep_tensor_inverse: fixes x :: "'b tensor" shows "Abs_tensor (Rep_tensor x) = x"
add_opp_double_opp: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c"   and p\<^sub>1 :: "'c point"   and p\<^sub>2 :: "'c point" assumes "ell_field R"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "ell_field.nonsingular R a b"   and "ell_field.on_curve R a b p\<^sub>1"   and "ell_field.on_curve R a b p\<^sub>2"   and "ell_field.add R a p\<^sub>1 p\<^sub>2 = ell_field.opp R p\<^sub>1" shows "p\<^sub>2 = ell_field.add R a (ell_field.opp R p\<^sub>1) (ell_field.opp R p\<^sub>1)"
steps_imp_chainp: fixes n :: "nat"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "nat \<Rightarrow> 'a" assumes "\<forall>i\<ge>n. \<exists>j>i. P (S i) (S j)" shows "\<exists>T. \<forall>i. P (T i) (T (Suc i))"
simps: fixes a :: "msg"   and H :: "msg set" shows "(a \<in> analz H) = ((\<exists>X. a = X \<and> X \<in> H) \<or> (\<exists>X Y. a = X \<and> \<langle>X, Y\<rangle> \<in> analz H) \<or> (\<exists>X Y. a = Y \<and> \<langle>X, Y\<rangle> \<in> analz H) \<or> (\<exists>X Y. a = X \<and> Enc X Y \<in> analz H \<and> Y \<in> synth (analz H)) \<or> (\<exists>X Y. a = X \<and> Aenc X (pubK Y) \<in> analz H \<and> priK Y \<in> analz H) \<or> (\<exists>X Y. a = X \<and> Aenc X (epubK Y) \<in> analz H \<and> epriK Y \<in> synth (analz H)) \<or> (\<exists>X Y. a = X \<and> Sign X (priK Y) \<in> analz H \<and> pubK Y \<in> analz H))"
fixp_induct: fixes Pa :: "('c llist \<Rightarrow> 'c llist) \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool" assumes "llist.admissible Pa"   and "Pa (\<lambda>lfilter. LNil)"   and "\<And>xs. \<lbrakk>Pa xs; llist.le_fun xs (lfilter P)\<rbrakk> \<Longrightarrow> Pa (\<lambda>xsa. case xsa of LNil \<Rightarrow> LNil | LCons x xs' \<Rightarrow> if P x then LCons x (xs xs') else xs xs')" shows "Pa (lfilter P)"
init_fin_FWdelay_bisimulation_final_base: shows "FWdelay_bisimulation_final_base r1.init_fin_final r1.init_fin r2.init_fin_final r2.init_fin init_fin_bisim r1.init_fin_\<tau>move r2.init_fin_\<tau>move"
top_closed: shows "\<top>\<^bsub>fpl L f\<^esub> \<in> carrier (fpl L f)"
rbt_update: fixes t :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and x :: "'c"   and y :: "'d" assumes "rbt t" shows "rbt (update x y t)"
steps_preserves_termination: fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and c :: "('a, 'b, 'c) com"   and s :: "('a, 'c) xstate"   and c' :: "('a, 'b, 'c) com"   and s' :: "('a, 'c) xstate" assumes "\<Gamma>\<turnstile> (c, s) \<rightarrow>\<^sup>* (c', s')"   and "\<Gamma>\<turnstile>c \<down> s" shows "\<Gamma>\<turnstile>c' \<down> s'"
simps_4: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b list" shows "map_iarray f (IArray x) = IArray (map f x)"
Gromov_extension_def: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d Gromov_completion" shows "Gromov_extension f x = (if x \<in> Gromov_boundary then lim (to_Gromov_completion \<circ> f \<circ> rep_Gromov_completion x) else to_Gromov_completion (f (from_Gromov_completion x)))"
\<tau>multithreaded_wf_init_fin: shows "\<tau>multithreaded_wf r1.init_fin_final r1.init_fin r1.init_fin_\<tau>move"
homotopic_with_trans: fixes P :: "('b \<Rightarrow> 'c) \<Rightarrow> bool"   and X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "homotopic_with P X Y f g"   and "homotopic_with P X Y g h" shows "homotopic_with P X Y f h"
map_FGcontra_def: fixes l1 :: "'p \<Rightarrow> 'ac"   and co1 :: "'q \<Rightarrow> 'ad"   and co2 :: "'y \<Rightarrow> 'al"   and co3 :: "'r \<Rightarrow> 'ae"   and co4 :: "'s \<Rightarrow> 'af"   and co5 :: "'t \<Rightarrow> 'ag"   and contra1 :: "'ah \<Rightarrow> 'u"   and contra2 :: "'ai \<Rightarrow> 'v"   and contra3 :: "'aj \<Rightarrow> 'w"   and contra4 :: "'ak \<Rightarrow> 'x"   and contra5 :: "'am \<Rightarrow> 'aa" shows "map_FGcontra l1 co1 co2 co3 co4 co5 contra1 contra2 contra3 contra4 contra5 = map_F l1 co1 co3 co1 co4 co5 (map_G contra1 contra2 contra3 contra4 co1 co2) contra1 contra5"
phiArg_same_var: fixes g :: "'g"   and p :: "'val"   and q :: "'val" assumes "phiArg g p q" shows "var g q = var g p"
obtain_sorted_carrier'_def: shows "obtain_sorted_carrier' \<equiv> SPEC (\<lambda>L. wsorted' L \<and> \<alpha> ` set L = E)"
intros_5: shows "CK_nf_pos (K \<circ> I)"
ord_gpv_trans: fixes f :: "('d, 'e, 'f) gpv"   and g :: "('d, 'e, 'f) gpv"   and h :: "('d, 'e, 'f) gpv" assumes "ord_gpv f g"   and "ord_gpv g h" shows "ord_gpv f h"
spmf_chain_countable: shows "countable (\<Union> (set_spmf ` Y))"
neutralSeller: fixes a :: "(integer \<times> 'c) set"   and b :: "integer \<times> 'c \<Rightarrow> 'd" assumes "\<forall>X. X \<in> Range a \<longrightarrow> b (seller, X) = (0::'d)"   and "finite a" shows "sum b a = sum b (Outside' {seller} a)"
one_mod_two_eq_one: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "modulo one (numeral.numeral one plus (num.Bit0 num.One)) = one"
landau_function_family_axioms: shows "landau_function_family F H"
bij_betw_dcharacters_characters: shows "bij_betw dc2c (dcharacters n) (characters G)"
inf_not_le_Le: fixes x :: "'d" shows "(\<infinity> \<le> Le x) = False"
ltl_prop_implication_implies_ltl_implication: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltl"   and \<psi> :: "'a ltl" assumes "w \<Turnstile> \<phi>"   and "\<phi> \<longrightarrow>\<^sub>P \<psi>" shows "w \<Turnstile> \<psi>"
remove_nth_sound_l: fixes p :: "nat"   and n :: "nat"   and xs :: "'c list" assumes "p < n" shows "remove_nth n xs ! p = xs ! p"
geodesic_subsetI: fixes S :: "'b set" assumes "\<And>x y. \<lbrakk>x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> \<exists>G. geodesic_segment_between G x y \<and> G \<subseteq> S" shows "geodesic_subset S"
mod_ring_mult: shows "(mod_ring_rel ===> mod_ring_rel ===> mod_ring_rel) (mult_p p) (*)"
Re_interval_minus: fixes A :: "complex_interval"   and B :: "complex_interval" shows "Re_interval (A - B) = Re_interval A - Re_interval B"
rt_strictly_fresherE': fixes i :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and P :: "(nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option) \<Rightarrow> (nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option) \<Rightarrow> nat \<Rightarrow> bool" assumes "rt1 \<sqsubset>\<^bsub>i\<^esub> rt2"   and "\<lbrakk>rt1 \<sqsubseteq>\<^bsub>i\<^esub> rt2; \<not> rt2 \<sqsubseteq>\<^bsub>i\<^esub> rt1\<rbrakk> \<Longrightarrow> P rt1 rt2 i" shows "P rt1 rt2 i"
sel_4: fixes x1 :: "'a"   and x2 :: "'b"   and x3 :: "nat"   and x4 :: "('a, 'b) BinomialHeapStruc.BinomialTree list" shows "BinomialHeapStruc.children (BinomialHeapStruc.Node x1 x2 x3 x4) = x4"
run_catch_option: fixes return :: "'e option \<Rightarrow> 'd"   and bind :: "'d \<Rightarrow> ('e option \<Rightarrow> 'd) \<Rightarrow> 'd"   and m1 :: "('e, 'd) optionT"   and m2 :: "('e, 'd) optionT" shows "run_option (catch_option return bind m1 m2) = bind (run_option m1) (\<lambda>x. if x = None then run_option m2 else return x)"
summable_complex_of_real: fixes f :: "nat \<Rightarrow> real" shows "summable (\<lambda>n. complex_of_real (f n)) = summable f"
take_rews_1: fixes n :: "nat" shows "slist_take (Suc n)\<cdot>[::] = [::]"
iso_Euclidean_complements_lemma2: fixes m :: "nat"   and S :: "(nat \<Rightarrow> real) set"   and n :: "nat"   and T :: "(nat \<Rightarrow> real) set"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real" assumes "closedin (Euclidean_space m) S"   and "closedin (Euclidean_space n) T"   and "homeomorphic_map (subtopology (Euclidean_space m) S) (subtopology (Euclidean_space n) T) f" obtains g :: "(nat \<Rightarrow> real) \<times> (nat \<Rightarrow> real) \<Rightarrow> (nat \<Rightarrow> real) \<times> (nat \<Rightarrow> real)" where "homeomorphic_map (prod_topology (Euclidean_space m) (Euclidean_space n)) (prod_topology (Euclidean_space n) (Euclidean_space m)) g" and "\<And>x. x \<in> S \<Longrightarrow> g (x, \<lambda>i. 0) = (f x, \<lambda>i. 0)"
strategyproofness_1: shows "(pmf (sds R26) d + pmf (sds R26) c < pmf (sds R25) d + pmf (sds R25) b \<or> pmf (sds R26) d + (pmf (sds R26) c + pmf (sds R26) a) < pmf (sds R25) d + (pmf (sds R25) b + pmf (sds R25) a)) \<or> pmf (sds R26) d + pmf (sds R26) c = pmf (sds R25) d + pmf (sds R25) b \<and> pmf (sds R26) d + (pmf (sds R26) c + pmf (sds R26) a) = pmf (sds R25) d + (pmf (sds R25) b + pmf (sds R25) a)"
lexlist_less_def: fixes xs :: "'b lexlist"   and ys :: "'b lexlist" shows "(xs < ys) = list_less (unlex xs) (unlex ys)"
total: fixes x :: "'alt"   and y :: "'alt" assumes "x \<in> alts"   and "y \<in> alts" shows "R i x y \<or> R i y x"
applyBackjump_def: fixes state :: "State" shows "applyBackjump state = (let l = getCl state; level = getBackjumpLevel state; state' = state\<lparr>getConflictFlag := False, getQ := [], getM := prefixToLevel level (getM state)\<rparr> in Let (if 0 < level then setReason (opposite l) (length (getF state) - 1) state' else state') (assertLiteral (opposite l) False))"
size_Un_Int: fixes A :: "'a multiset"   and B :: "'a multiset" shows "size A + size B = size (A \<union># B) + size (A \<inter># B)"
is_group: shows "Group.group (DirProds (f \<circ> Gs) I)"
map_it_to_list_genord_correct: fixes it :: "'c \<Rightarrow> (('a \<times> 'b) list \<Rightarrow> bool) \<Rightarrow> ('a \<times> 'b \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('a \<times> 'b) list) \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('a \<times> 'b) list"   and s :: "'c"   and m :: "'a \<Rightarrow> 'b option"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "map_iterator_genord (it s) m (\<lambda>(k, uu_) (k', uu_). R k k')" shows "map_of (it_to_list it s) = m \<and> distinct (map fst (it_to_list it s)) \<and> sorted_wrt R (map fst (it_to_list it s))"
bigo_abs: fixes f :: "'c \<Rightarrow> 'd" shows "(\<lambda>x. \<bar>f x\<bar>) \<in> O(f)"
suc\<rho>: shows "\<rho> (Suc k) = \<pi> (Suc k)"
diff_commute: fixes i :: "nat"   and j :: "nat"   and k :: "nat" shows "i - j - k = i - k - j"
irreducible_mult_unit_left: fixes a :: "'a"   and p :: "'a" assumes "a dvd (1::'a)" shows "irreducible (a * p) = irreducible p"
odd_iff_mod_2_eq_one: 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 divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "(\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a) = (modulo a (numeral.numeral one plus (num.Bit0 num.One)) = one)"
power_one: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and n :: "nat" assumes "class.monoid_mult one times" shows "power.power one times one n = one"
r_into_trancl': fixes p :: "'e \<times> 'e"   and r :: "('e \<times> 'e) set" assumes "p \<in> r" shows "p \<in> r\<^sup>+"
empty_subseqs: fixes xs :: "'a list" shows "[] \<in> set (subseqs xs)"
complex_cnj_numeral: fixes w :: "num" shows "cnj (numeral w) = numeral w"
assms_9: shows "e\<^sup>T \<le> w\<^sup>\<star>"
reds_preserves_lconf: fixes es :: "char list exp list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and es' :: "char list exp list"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l' :: "char list \<Rightarrow> val option"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and E :: "char list \<Rightarrow> ty option"   and Ts :: "ty list" assumes "P \<turnstile> \<langle>es,(h, l)\<rangle> [\<rightarrow>] \<langle>es',(h', l')\<rangle>"   and "P,E,h \<turnstile> es [:] Ts"   and "P,h \<turnstile> l (:\<le>) E" shows "P,h' \<turnstile> l' (:\<le>) E"
conjcases: fixes P :: "bool"   and Q :: "bool" obtains "P \<and> Q"   | "P \<and> \<not> Q"   | "\<not> P \<and> Q"   | "\<not> P \<and> \<not> Q"
prems: fixes x :: "nat" assumes "x \<in> set (x__ # xs) - set ys" shows "\<exists>y. y \<in> set ys \<and> x \<le> y"
HCond_Cont: fixes hcond :: "'a hrel \<Rightarrow> 'a hrel" assumes "utp_theory_continuous hcond" shows "Continuous hcond"
Derives1_X_is_part_of_rule: fixes \<delta> :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and a :: "symbol list"   and X :: "symbol"   and b :: "symbol list"   and \<alpha> :: "symbol list"   and N :: "symbol"   and \<beta> :: "symbol list" assumes "Derives1 \<delta> i r (a @ [X] @ b)"   and "splits_at \<delta> i \<alpha> N \<beta>"   and "\<And>\<beta>. \<lbrakk>\<delta> = a @ [X] @ \<beta>; length a < i; Derives1 \<beta> (i - length a - 1) r b\<rbrakk> \<Longrightarrow> False"   and "\<And>\<alpha>. \<lbrakk>\<delta> = \<alpha> @ [X] @ b; Derives1 \<alpha> i r a\<rbrakk> \<Longrightarrow> False" shows "\<exists>u v. a = \<alpha> @ u \<and> b = v @ \<beta> \<and> snd r = u @ [X] @ v"
odiff_add_cancel_both: fixes x :: "V"   and y :: "V"   and z :: "V" shows "odiff (x + y) (x + z) = odiff y z"
abs_triangle_ineq: fixes abs :: "'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq (abs (plus a b)) (plus (abs a) (abs b))"
f_1: shows "range f \<subseteq> emeasure M ` {B \<in> sets M. A \<subseteq> B}"
RModuleHom_axioms: shows "RModuleHom UNIV (\<cdot>) M (\<cdot>) T"
sturm_indices: fixes i :: "nat"   and p :: "real poly" assumes "i < length (sturm p) - 2" shows "sturm p ! (i + 2) = - (sturm p ! i mod sturm p ! (i + 1))"
lt: shows "\<lhd>\<^sub>u\<^sub>b (lt dir) c (UB dir s x)"
1: shows "map ((!) (as @ bs)) xs = map ((!) as) xs"
ID_unfold_vars: fixes x :: "'a"   and y :: "'a"   and T :: "'b itself" assumes "ID x y T" shows "x \<equiv> y"
snap_p_2: shows "ps (S t (i + 1)) p \<noteq> None"
subseqs_set: fixes y :: "'a list"   and xs :: "'a list" assumes "y \<in> set (subseqs xs)" shows "set y \<subseteq> set xs"
get_nondet_def: fixes get :: "('f \<Rightarrow> 'e) \<Rightarrow> 'e"   and f :: "'f \<Rightarrow> ('d, 'e) nondetT" shows "nondetM_base.get_nondet get f = NondetT (get (\<lambda>s. run_nondet (f s)))"
WT_gpv_mk_lossless_gpv: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and \<I>' :: "('a, 'b) \<I>"   and x :: "'c" assumes "\<I> \<turnstile>g gpv \<surd>"   and "outs_\<I> \<I>' = outs_\<I> \<I>" shows "\<I>' \<turnstile>g mk_lossless_gpv (responses_\<I> \<I>) x gpv \<surd>"
monad_prob_envT: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and sample :: "'c pmf \<Rightarrow> ('c \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_prob return bind sample" shows "monad_prob (return_env return) (bind_env bind) (sample_env sample)"
assms_1: shows "imp \<phi> \<psi> \<pi>l"
state_update_vs_allInstances_generic_including': fixes pre_post :: "'d state \<times> 'd state \<Rightarrow> 'd state"   and mk :: "'d state \<Rightarrow> 'd state \<times> 'd state"   and \<sigma>' :: "nat \<Rightarrow> \<langle>'d\<rangle>\<^sub>\<bottom>"   and oid :: "nat"   and Object :: "'d"   and Type :: "'d \<Rightarrow> \<langle>'e\<rangle>\<^sub>\<bottom>"   and A :: "nat \<Rightarrow> \<langle>nat list list list\<rangle>\<^sub>\<bottom>" assumes "\<And>a. pre_post (mk a) = a"   and "\<And>x. \<sigma>' oid = \<lfloor>x\<rfloor> \<Longrightarrow> x = Object"   and "Type Object \<noteq> None" shows "OclAllInstances_generic pre_post Type (mk \<lparr>heap = \<sigma>'(oid \<mapsto> Object), assocs = A\<rparr>) = OclAllInstances_generic pre_post Type->including\<^sub>S\<^sub>e\<^sub>t(\<lambda>_. \<lfloor>\<lfloor>\<lceil>Type Object\<rceil>\<rfloor>\<rfloor>) (mk \<lparr>heap = \<sigma>', assocs = A\<rparr>)"
reindex_bij_betw: fixes h :: "'b \<Rightarrow> 'c"   and S :: "'b set"   and T :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "bij_betw h S T" shows "(\<Sum>x\<in>S. g (h x)) = sum g T"
up_dense_stone_phi: fixes x :: "'c" shows "up_filter (Abs_dense (x \<squnion> - x)) \<le> stone_phi (Abs_regular (- - x))"
map_add_None: fixes m :: "'d \<Rightarrow> 'c option"   and n :: "'d \<Rightarrow> 'c option"   and k :: "'d" shows "((m ++ n) k = None) = (n k = None \<and> m k = None)"
sur_total: fixes x :: "'a" shows "is_sur x = is_total (x\<^sup>\<smile>)"
simps_2: fixes n :: "nat" shows "FV (ENat n) = {}"
closed_path_image: fixes g :: "real \<Rightarrow> 'b" assumes "path g" shows "closed (path_image g)"
bool_assn_proper_2: shows "proper (\<lambda>_. False)"
r_zero: fixes x :: "'e" assumes "x \<in> carrier N" shows "x \<oplus>\<^bsub>N\<^esub> \<zero>\<^bsub>N\<^esub> = x"
mat_max_comm: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'a mat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc" shows "max\<^sub>m A B = max\<^sub>m B A"
x: shows "x \<in> B\<^sub>2 - B\<^sub>1"
lookup_eq: shows "Aw' $$ (i, j) = lookup Aw is"
esing_below_iff: fixes x :: "'c"   and a :: "'d"   and ae :: "'c \<Rightarrow> 'd" shows "(esing x\<cdot>a \<sqsubseteq> ae) = (a \<sqsubseteq> ae x)"
1: fixes u :: "real"   and v :: "real" assumes "u + v = 1" shows "u *\<^sub>R a + v *\<^sub>R b = a + v *\<^sub>R (b - a)"
rep_eq: shows "\<langle>\<lbrace>\<rbrace>\<rangle>\<^sub>f = {}"
class_add_wt_method: fixes P :: "jvm_prog"   and C\<^sub>0 :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T\<^sub>r :: "ty"   and mxs :: "nat"   and mxl\<^sub>0 :: "nat"   and "is" :: "bytecode"   and xt :: "ex_table"   and \<Phi> :: "char list \<Rightarrow> 'a \<Rightarrow> (ty list \<times> ty err list) option list"   and M\<^sub>0 :: "'a"   and wf_md :: "jvm_prog \<Rightarrow> char list \<Rightarrow> jvm_method mdecl \<Rightarrow> bool"   and C :: "char list"   and cdec :: "jvm_method class" assumes "wt_method P C\<^sub>0 b Ts T\<^sub>r mxs mxl\<^sub>0 is xt (\<Phi> C\<^sub>0 M\<^sub>0)"   and "wf_prog wf_md P"   and "\<not> is_class P C" shows "wt_method (class_add P (C, cdec)) C\<^sub>0 b Ts T\<^sub>r mxs mxl\<^sub>0 is xt (\<Phi> C\<^sub>0 M\<^sub>0)"
int_pow_0: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" shows "x [^]\<^bsub>G\<^esub> 0 = \<one>\<^bsub>G\<^esub>"
eq: shows "(\<Sum>i\<le>Suc b. monom (g i) i) = monom (g (Suc b)) (Suc b) \<oplus>\<^bsub>R\<^esub> (\<Sum>i\<le>b. monom (g i) i)"
trans_WhileFalse: fixes tst :: "'test"   and s :: "'state"   and c :: "('test, 'atom) com" assumes "\<not> tval tst s" shows "(while tst do c, s) \<rightarrow>t s"