hcju/isabelleps huggingface dataset:
mbind_nofailure: fixes S :: "'f list"   and f :: "'f \<Rightarrow> 'e \<Rightarrow> ('d \<times> 'e) option"   and \<sigma> :: "'e" shows "mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>S\<^sub>a\<^sub>v\<^sub>e S f \<sigma> \<noteq> None"
set_cases: fixes e :: "'e"   and a :: "'e list" assumes "e \<in> set a" obtains z2 :: "'e list" where "a = Cons e z2__"   | z1 :: "'e" and z2 :: "'e list" where "a = Cons z1 z2" and "e \<in> set z2"
exhaust: fixes y :: "('c, 'd) bintree" obtains x1 :: "'c" and x2 :: "'d" and x3 :: "('c, 'd) bintree list" where "y = Node x1 x2 x3"
29: fixes x :: "'a" shows "bot \<squnion> x = x"
assms_5: shows "p = real (card A) / real (card A + card B)"
drop_i': shows "snd (splitAt x xs) = drop (Suc i) xs"
n_2: shows "Sigma_Algebra.measure M (T ((1 / 2) ^ n)) < Sigma_Algebra.measure M S + epsilon"
fds_ln_mult: fixes f :: "'b fds"   and g :: "'b fds"   and l' :: "'b"   and l'' :: "'b"   and l :: "'b" assumes "fds_nth f 1 \<noteq> (0::'b)"   and "fds_nth g 1 \<noteq> (0::'b)"   and "l' + l'' = l" shows "fds_ln l (f * g) = fds_ln l' f + fds_ln l'' g"
in_ocircline_cmat_cvec_def: fixes H :: "complex \<times> complex \<times> complex \<times> complex"   and z :: "complex \<times> complex" shows "in_ocircline_cmat_cvec H z = (Re (quad_form z H) < 0)"
CConcStart2: fixes p :: "((int \<times> int) set set, int) Combinators list"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and a :: "((int \<times> int) set set, int) Combinators" assumes "p \<noteq> []"   and "x \<notin> dom (C a)" shows "C (list2FWpolicy (p @ [a])) x = C (list2FWpolicy p) x"
upper_asymptotic_densityD: fixes A :: "nat set"   and l :: "real" assumes "upper_asymptotic_density A < l" shows "\<forall>\<^sub>F n in sequentially. real (card (A \<inter> {..<n})) < l * real n"
half_gt_zero_iff: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less zero (divide a (numeral.numeral one plus (num.Bit0 num.One))) = less zero a"
owner_document: shows "h \<turnstile> get_owner_document ptr \<rightarrow>\<^sub>r owner_document"
ide_compE: fixes g :: "'b"   and f :: "'b" assumes "D.ide (g \<cdot>\<^sub>D f)" obtains "D.seq g f" and "D.seq f g" and "g \<cdot>\<^sub>D f = D.dom f" and "g \<cdot>\<^sub>D f = D.cod g"
Decrypt_def: shows "Decrypt \<equiv> (id ---> rep_msg ---> abs_msg) DECRYPT"
check_not_equal_dom_err_def: fixes p :: "(char list list \<times> int) list"   and q :: "(char list list \<times> int) list"   and pq :: "(char list list \<times> int) list"   and r :: "(char list list \<times> int) list" shows "check_not_equal_dom_err p q pq r = SPEC (\<lambda>_. True)"
J: shows "offs ns 0 ! Max {i. i < length ns \<and> offs ns 0 ! i < k} = foldl (+) 0 (take (Max {i. i < length ns \<and> offs ns 0 ! i < k}) ns)"
lessThan_eq_iff: fixes x :: "'b"   and y :: "'b" shows "({..<x} = {..<y}) = (x = y)"
path_shift_set_cd: fixes \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat"   and n :: "nat" assumes "is_path \<pi>" shows "{k + j |j. n cd\<^bsup>\<pi> \<guillemotleft> k\<^esup>\<rightarrow> j} = {i. k + n cd\<^bsup>\<pi>\<^esup>\<rightarrow> i \<and> k \<le> i}"
sumle: shows "\<forall>n. matrix_sum d (\<lambda>n. denote_while_n M0 M1 DS n \<rho>) n \<le>\<^sub>L matrix_sum d (\<lambda>n. denote_while_n M0 M1 DS n \<rho>) (Suc n)"
disjoint_iff: fixes A :: "'f set"   and B :: "'f set" shows "(A \<inter> B = {}) = (\<forall>x. x \<in> A \<longrightarrow> x \<notin> B)"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'c tree23 \<Rightarrow> 'b"   and f2 :: "'c tree23 \<Rightarrow> 'c \<Rightarrow> 'c tree23 \<Rightarrow> 'b"   and upI :: "'c upI" shows "P (case upI of TI x \<Rightarrow> f1 x | OF x xa xb \<Rightarrow> f2 x xa xb) = ((\<forall>x1. upI = TI x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x21 x22 x23. upI = OF x21 x22 x23 \<longrightarrow> P (f2 x21 x22 x23)))"
elims: fixes x :: "val"   and y :: "ty" assumes "type x = y" obtains i :: "int" where "x = Iv i" and "y = Ity"   | r :: "real" where "x = Rv r" and "y = Rty"
sup_hf_def: fixes a :: "hf"   and b :: "hf" shows "a \<squnion> b = (THE z. \<forall>u. (u \<^bold>\<in> z) = (u \<^bold>\<in> a \<or> u \<^bold>\<in> b))"
quant_equiv_iff: fixes x :: "'e['f, 'g]"   and y :: "'e['h, 'g]" shows "(x \<cong>\<^sub>Q y) = (\<lbrakk>x\<rbrakk>\<^sub>Q = \<lbrakk>y\<rbrakk>\<^sub>Q \<and> QD('f) = QD('h))"
card_farey_fractions_asymptotics: shows "(\<lambda>N. real (card (farey_fractions N)) - 3 / pi\<^sup>2 * (real N)\<^sup>2) \<in> O(\<lambda>N. real N * ln (real N))"
frrestriction_flrestriction: fixes r :: "V"   and A :: "V" shows "(r \<restriction>\<^sup>r\<^sub>\<bullet> A) \<restriction>\<^sup>l\<^sub>\<bullet> A = r \<restriction>\<^sub>\<bullet> A"
funct_OT_12_def: fixes input\<^sub>1 :: "'a \<times> 'a"   and \<sigma> :: "bool" shows "funct_OT_12 input\<^sub>1 \<sigma> = return_spmf ((), if \<sigma> then snd input\<^sub>1 else fst input\<^sub>1)"
that: fixes Q' :: "'state" assumes "Q \<Rightarrow> Q'"   and "P' \<approx>\<cdot> Q'" shows "thesis"
uniformly_continuous_on_Cauchy: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and X :: "nat \<Rightarrow> 'b" assumes "uniformly_continuous_on S f"   and "Cauchy X"   and "\<And>n. X n \<in> S" shows "Cauchy (\<lambda>n. f (X n))"
combine_conditions_on_relations_indRelR_1: fixes RelA :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and CondA :: "('procS, 'procT) Proc \<times> ('procS, 'procT) Proc \<Rightarrow> bool"   and RelB :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and CondB :: "('procS, 'procT) Proc \<times> ('procS, 'procT) Proc \<Rightarrow> bool" assumes "\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> RelA"   and "\<forall>(P, Q)\<in>RelA. CondA (P, Q)"   and "\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> RelB"   and "\<forall>(P, Q)\<in>RelB. CondB (P, Q)" shows "\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> (\<forall>(P, Q)\<in>Rel. CondA (P, Q)) \<and> (\<forall>(P, Q)\<in>Rel. CondB (P, Q))"
valid_def: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and n :: "nat" shows "valid P c Q n = (\<forall>s. P s \<longrightarrow> (\<exists>s' m. (c, s) \<Rightarrow> m \<Down> s' \<and> m \<le> n \<and> Q s'))"
proj_one: fixes t :: "cnf llist"   and c :: "'id" assumes "\<exists>i. enat i < llength t \<and> \<parallel>c\<parallel>\<^bsub>lnth t i\<^esub>" shows "1 \<le> llength (\<pi>\<^bsub>c\<^esub>t)"
measurable_T2: fixes M :: "'c measure"   and K :: "'d \<Rightarrow> 'd pmf"   and s :: "'d" shows "M \<rightarrow>\<^sub>M MC_syntax.T K s = M \<rightarrow>\<^sub>M stream_space (count_space UNIV)"
hom_uminus: fixes x :: "int poly" shows "of_int_poly (- x) = - of_int_poly x"
insert_3: assumes "\<And>x. x \<in> S \<Longrightarrow> set (coeffs (f x)) \<subseteq> \<real>" shows "set (coeffs (sum f S)) \<subseteq> \<real>"
whileLoop_def: fixes C :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> ('c \<times> 'd) set \<times> bool" shows "whileLoop C   B \<equiv> \<lambda>r s. ({(r', s'). (Some (r, s), Some (r', s')) \<in> whileLoop_results C B}, (Some (r, s), None) \<in> whileLoop_results C B \<or> \<not> whileLoop_terminates C B r s)"
is_isoI: fixes h :: "'d \<Rightarrow> 'e"   and G :: "('d, 'f) monoid_scheme"   and H :: "('e, 'g) monoid_scheme" assumes "h \<in> Group.iso G H" shows "G \<cong> H"
in_set_conv_nth: fixes x :: "'a"   and xs :: "'a list" shows "(x \<in> set xs) = (\<exists>i<length xs. xs ! i = x)"
intros_2: fixes A :: "agent"   and H :: "msg set" shows "Messages.Agent A \<in> synth H"
positively_invariant_restrict_dom: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and M :: "'b set" assumes "auto_ll_on_open f X" shows "auto_ll_on_open.positively_invariant f X M = auto_ll_on_open.positively_invariant f X (M \<inter> X)"
epi_def: fixes G :: "('e, 'g) monoid_scheme"   and H :: "('f, 'h) monoid_scheme" shows "epi G H = {f \<in> hom G H. f ` carrier G = carrier H}"
If: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and b :: "bexp"   and c\<^sub>1 :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c\<^sub>2 :: "com" assumes "\<turnstile>\<^sub>2 {\<lambda>s. P s + \<up> (bval b s)} c\<^sub>1 {Q}"   and "\<turnstile>\<^sub>2 {\<lambda>s. P s + \<up> (\<not> bval b s)} c\<^sub>2 {Q}" shows "\<turnstile>\<^sub>2 {\<lambda>s. eSuc (P s)} IF b THEN c\<^sub>1 ELSE c\<^sub>2 {Q}"
restrict_extensional: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" shows "restrict f A \<in> extensional A"
cases: fixes x :: "'v node \<times> 'v node" obtains xa :: "'v" where "x = (SOURCE, Inner xa__)"   | xa :: "'v" and y :: "'v" where "x = (Inner xa, Inner y)"   | vb :: "'v" where "x = (Inner vb, SOURCE)"   | v :: "'v node" where "x = (v, SOURCE)"
sameDom_trans: fixes inp :: "'d \<Rightarrow> 'e option"   and inp' :: "'d \<Rightarrow> 'f option"   and inp'' :: "'d \<Rightarrow> 'g option" assumes "sameDom inp inp'"   and "sameDom inp' inp''" shows "sameDom inp inp''"
knows_composition2_k: fixes m :: "Keys"   and P :: "specID"   and Q :: "specID"   and PQ :: "specID" assumes "kKS m \<notin> specKeysSecrets P"   and "kKS m \<notin> specKeysSecrets Q"   and "knows Q [kE m]"   and "subcomponents PQ = {P, Q}"   and "correctCompositionIn PQ"   and "correctCompositionKS PQ" shows "knows PQ [kE m]"
twoCisTypeII_7: shows "(\<lambda>x. twoC (0, x)) = (\<lambda>x. (g2 (a + (b - a) * x), a + (b - a) * x))"
a1: shows "(\<lambda>i. map_option (fst \<circ> sr) (netmap s1 i)) = fst (netgmap sr s1)"
to_bl_to_bin: fixes w :: "'b word" shows "bl_to_bin (to_bl w) = uint w"
functor_axioms: shows "functor (\<cdot>) \<F>\<^sub>SC.comp D"
insert_compr: fixes a :: "'e"   and B :: "'e set" shows "insert a B = {x. x = a \<or> x \<in> B}"
node_deliverTE'': fixes ii :: "nat"   and s :: "'c"   and R :: "nat set"   and i :: "nat"   and d :: "nat"   and s' :: "'c net_state"   and S :: "('c \<times> 'd seq_action \<times> 'c) set" assumes "(NodeS ii s R, i:deliver(d), s') \<in> node_sos S" obtains s'a :: "'c" where "i = ii" and "s' = NodeS ii s'a R" and "(s, deliver d, s'a) \<in> S"
id_3: shows "length (bef @ s # aft) = Suc (length bef + length aft)"
map_id: fixes t :: "'c \<Rightarrow> 'd" shows "id \<circ> t = t"
fold_A_chamber_images_2: shows "fold_A ` D = C"
disc_iff_1: fixes n :: "enat"   and xs :: "'c llist" shows "lnull (ltake n xs) = (n = 0 \<or> lnull xs)"
principal_ideal_n_pow: fixes a :: "'a"   and I :: "'a set"   and n :: "nat" assumes "a \<in> carrier R"   and "I = R \<diamondsuit>\<^sub>p a" shows "I \<^bsup>\<diamondsuit>R n\<^esup> = R \<diamondsuit>\<^sub>p (a^\<^bsup>R n\<^esup>)"
parent_child_rel_finite: fixes h :: "(_) heap" assumes "heap_is_wellformed h" shows "finite (parent_child_rel h)"
approximating_bigstep_fun_seq_Undecided_t_wf: fixes \<gamma> :: "('a \<Rightarrow> 'b \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'b \<Rightarrow> bool)"   and p :: "'b"   and rs1 :: "'a rule list"   and rs2 :: "'a rule list"   and t :: "state" assumes "wf_ruleset \<gamma> p (rs1 @ rs2)" shows "(approximating_bigstep_fun \<gamma> p (rs1 @ rs2) Undecided = t) = (approximating_bigstep_fun \<gamma> p rs1 Undecided = Undecided \<and> approximating_bigstep_fun \<gamma> p rs2 Undecided = t \<or> approximating_bigstep_fun \<gamma> p rs1 Undecided = t \<and> t \<noteq> Undecided)"
plus_pp_pp: fixes x :: "('a, 'b) pp"   and y :: "('a, 'b) pp" shows "rep_nat_pp (x + y) = rep_nat_pp x + rep_nat_pp y"
finite_infinity: fixes a :: "('a, 'b) IEEE.float" assumes "is_finite a" shows "\<not> is_infinity a"
cases: fixes x :: "'b formula" obtains k :: "'b" where "x = Atom k"   | "x = \<bottom>"   | F :: "'b formula" where "x = \<^bold>\<not> Fa__"   | F :: "'b formula" and G :: "'b formula" where "x = Fb__ \<^bold>\<and> G__"   | F :: "'b formula" and G :: "'b formula" where "x = Fc__ \<^bold>\<or> Ga__"   | F :: "'b formula" and G :: "'b formula" where "x = F \<^bold>\<rightarrow> G"
node_ptr_kinds_eq_h: shows "node_ptr_kinds h2 = node_ptr_kinds h |\<union>| {|cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr|}"
ord_dvd_group_order: fixes a :: "nat \<Rightarrow> int" assumes "a \<in> carrier (DirProds (\<lambda>p. Z (p ^ multiplicity p n)) (prime_factors n))" shows "DP.ord a dvd order (DirProds (\<lambda>p. Z (p ^ multiplicity p n)) (prime_factors n))"
inv_imps_work_sum_zcount_def: fixes c :: "('loc, 't) configuration" shows "inv_imps_work_sum_zcount c \<equiv> \<forall>loc t. zcount (c_imp c loc + c_work c loc) t = zcount (zmset_of (mset_set (set_antichain (frontier (c_pts c loc)))) + (\<Sum>loc'\<in>UNIV. after_summary (zmset_of (mset_set (set_antichain (frontier (c_imp c loc'))))) (summary loc' loc))) t"
distinct_disc_18: fixes event :: "('d, 'e, 'f) event" assumes "isRecvMarker event" shows "\<not> isRecv event"
add_cancel_right_right: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.cancel_comm_monoid_add plus minus zero" shows "(a = plus a b) = (b = zero)"
Cons_1: assumes "distinct a'"   and "x \<in> set a'" shows "mset a' = {#x#} + mset (removeAll x a')"
Max_subset: fixes A :: "'b set"   and B :: "'b set" assumes "A \<noteq> {}"   and "A \<subseteq> B"   and "finite B" shows "Max A \<le> Max B"
conf_widen: fixes G :: "prog"   and T :: "ty"   and T' :: "ty"   and s :: "st"   and x :: "val" assumes "G\<turnstile>T\<preceq>T'"   and "G,s\<turnstile>x\<Colon>\<preceq>T"   and "ws_prog G" shows "G,s\<turnstile>x\<Colon>\<preceq>T'"
index_mult_mat_vec: fixes i :: "nat"   and A :: "'b mat"   and v :: "'b Matrix.vec" assumes "i < dim_row A" shows "(A *\<^sub>v v) $v i = Matrix.row A i \<bullet> v"
foldl_Cons: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and a :: "'b"   and x :: "'c"   and xs :: "'c list" shows "foldl f a (x # xs) = foldl f (f a x) xs"
healthy_monoD: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "healthy t" shows "mono_trans t"
simps: fixes st :: "'s \<times> ('c \<Rightarrow> real) set"   and a :: "('s \<times> ('c \<Rightarrow> real) set) pmf" shows "\<K>p st a = ((\<exists>l R R'. a = return_pmf (l, R') \<and> st \<in> \<S> \<and> st = (l, R) \<and> R' \<in> Succ \<R> R \<and> R' \<subseteq> {v. v \<turnstile> PTA.inv_of A l}) \<or> (\<exists>l R g \<mu>. a = map_pmf (\<lambda>(X, l). (l, [X::=0]R)) \<mu> \<and> st \<in> \<S> \<and> st = (l, R) \<and> (l, g, \<mu>) \<in> PTA.trans_of A \<and> R \<subseteq> {v. v \<turnstile> g}) \<or> a = return_pmf st)"
mergeFrameRes1: fixes A\<^sub>F :: "name list"   and \<Psi>\<^sub>G :: "'b"   and A\<^sub>G :: "name list"   and x :: "name"   and \<Psi>\<^sub>F :: "'b" assumes "A\<^sub>F \<sharp>* \<Psi>\<^sub>G"   and "A\<^sub>F \<sharp>* A\<^sub>G"   and "x \<sharp> A\<^sub>F"   and "x \<sharp> \<Psi>\<^sub>F"   and "A\<^sub>G \<sharp>* \<Psi>\<^sub>F" shows "(\<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle>) \<otimes>\<^sub>F \<lparr>\<nu>x\<rparr>\<langle>A\<^sub>G, \<Psi>\<^sub>G\<rangle> = \<langle>(A\<^sub>F @ x # A\<^sub>G), \<Psi>\<^sub>F \<otimes> \<Psi>\<^sub>G\<rangle>"
chamber_system_simplices: shows "order.greater_eq X folding_g.\<C>"
true_clss_set_mset: fixes I :: "'b set"   and CC :: "'b literal multiset multiset" shows "(I \<Turnstile>s set_mset CC) = (I \<Turnstile>m CC)"
closed_segment_idem: fixes a :: "'a" shows "closed_segment a a = {a}"
card_g1: shows "(k1, k2) \<in> cards s g1"
cases: fixes a1 :: "'a trm"   and a2 :: "'a trm" assumes "a1 \<rightarrow>\<beta> a2" obtains x :: "'a" and T :: "PreSimplyTyped.type" and A :: "'a trm" and M :: "'a trm" where "a1 = App (Fn x__ T__ A__) M" and "a2 = A__[x__ ::= M]"   | A :: "'a trm" and A' :: "'a trm" and B :: "'a trm" where "a1 = App Aa__ B__" and "a2 = App A'__ B__" and "Aa__ \<rightarrow>\<beta> A'__"   | B :: "'a trm" and B' :: "'a trm" and A :: "'a trm" where "a1 = App Ab__ Ba__" and "a2 = App Ab__ B'__" and "Ba__ \<rightarrow>\<beta> B'__"   | A :: "'a trm" and A' :: "'a trm" and x :: "'a" and T :: "PreSimplyTyped.type" where "a1 = Fn x T Ac__" and "a2 = Fn x T A'a__" and "Ac__ \<rightarrow>\<beta> A'a__"   | A :: "'a trm" and A' :: "'a trm" and B :: "'a trm" where "a1 = SimplyTyped.Pair Ad__ Bb__" and "a2 = SimplyTyped.Pair A' Bb__" and "Ad__ \<rightarrow>\<beta> A'"   | B :: "'a trm" and B' :: "'a trm" and A :: "'a trm" where "a1 = SimplyTyped.Pair Ae__ Bc__" and "a2 = SimplyTyped.Pair Ae__ B'" and "Bc__ \<rightarrow>\<beta> B'"   | P :: "'a trm" and P' :: "'a trm" where "a1 = Fst P__" and "a2 = Fst P'__" and "P__ \<rightarrow>\<beta> P'__"   | A :: "'a trm" and B :: "'a trm" where "a1 = Fst (SimplyTyped.Pair Af__ Bd__)" and "a2 = Af__"   | P :: "'a trm" and P' :: "'a trm" where "a1 = Snd P" and "a2 = Snd P'" and "P \<rightarrow>\<beta> P'"   | A :: "'a trm" and B :: "'a trm" where "a1 = Snd (SimplyTyped.Pair A B)" and "a2 = B"
compare_self_neq_GT: fixes x :: "'a" shows "compare\<cdot>x\<cdot>x \<noteq> GT"
oInv_uniq: fixes F :: "ordinal \<Rightarrow> ordinal"   and x :: "ordinal"   and a :: "ordinal"   and y :: "ordinal" assumes "mono F"   and "F x \<le> a \<and> a < F (oSuc x)"   and "F y \<le> a \<and> a < F (oSuc y)" shows "x = y"
prefix_eq_exists: fixes s :: "'b list"   and t :: "'b list" shows "prefix s t = (\<exists>xs. s @ xs = t)"
conv_has_start_points_path: fixes x :: "'a" shows "has_start_points_path x = has_end_points_path (x\<^sup>T)"
represent_NatInterval: fixes x :: "ZF"   and n :: "nat"   and m :: "nat" assumes "Elem x (NatInterval n m)" shows "\<exists>u\<ge>n. u \<le> n + m \<and> nat2Nat u = x"
db\<^sub>s\<^sub>s\<^sub>t_no_upd_append: fixes B :: "('a, 'b) stateful_strand_step list"   and A :: "('a, 'b) stateful_strand_step list" assumes "\<forall>b\<in>set B. \<not> is_Insert b \<and> \<not> is_Delete b" shows "db'\<^sub>s\<^sub>s\<^sub>t A = db'\<^sub>s\<^sub>s\<^sub>t (A @ B)"
is_arg_min_on_def: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and x :: "'c" shows "is_arg_min_on f S x = (x \<in> S \<and> (\<forall>y\<in>S. f x \<le> f y))"
sup_aci_3: fixes sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.semilattice_sup sup less_eq less" shows "sup x (sup y z) = sup y (sup x z)"
Unf\<^sub>G_\<G>: fixes \<G> :: "'b ltl set"   and \<phi> :: "'b ltl" assumes "Only_G \<G>"   and "\<G> \<Turnstile>\<^sub>P \<phi>" shows "\<G> \<Turnstile>\<^sub>P Unf\<^sub>G \<phi>"
map_eq_append_conv: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list"   and ys :: "'c list"   and zs :: "'c list" shows "(map f xs = ys @ zs) = (\<exists>us vs. xs = us @ vs \<and> ys = map f us \<and> zs = map f vs)"
zero_zhmultiset_def: shows "0 \<equiv> ZHMSet {#}\<^sub>z"
subspace_def: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set" assumes "module scale" shows "module.subspace scale S = ((0::'b) \<in> S \<and> (\<forall>x\<in>S. \<forall>y\<in>S. x + y \<in> S) \<and> (\<forall>c. \<forall>x\<in>S. scale c x \<in> S))"
split_case: fixes t :: "('b \<times> nat) tree" shows "split t = (case t of \<langle>\<rangle> \<Rightarrow> \<langle>\<rangle> | \<langle>t1, (x, lvx), \<langle>\<rangle>\<rangle> \<Rightarrow> \<langle>t1, (x, lvx), \<langle>\<rangle>\<rangle> | \<langle>t1, (x, lvx), \<langle>t2, (y, lvy), \<langle>\<rangle>\<rangle>\<rangle> \<Rightarrow> \<langle>t1, (x, lvx), \<langle>t2, (y, lvy), \<langle>\<rangle>\<rangle>\<rangle> | \<langle>t1, (x, lvx), \<langle>t2, (y, lvy), \<langle>t3, (z, lvz), t4\<rangle>\<rangle>\<rangle> \<Rightarrow> if lvx = lvy \<and> lvy = lvz then \<langle>\<langle>t1, (x, lvx), t2\<rangle>, (y, lvx + 1), \<langle>t3, (z, lvx), t4\<rangle>\<rangle> else t)"
le: fixes A :: "'a set" shows "measure_pmf.prob p A \<le> measure_pmf.prob q {y. \<exists>x\<in>A. R x y}"
measurable_sets_Collect: fixes f :: "'d \<Rightarrow> 'e"   and M :: "'d measure"   and N :: "'e measure"   and P :: "'e \<Rightarrow> bool" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "{x \<in> space N. P x} \<in> sets N" shows "{x \<in> space M. P (f x)} \<in> sets M"
TAQ_def: fixes x :: "('a, 'n) vec" shows "TAQ x = A ** Q *v x"
fresh_finite_insert: fixes S :: "'a set"   and a :: "atom"   and x :: "'a" assumes "finite S" shows "a \<sharp> insert x S = (a \<sharp> x \<and> a \<sharp> S)"
Derives1_bound: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and a :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and b :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "CFG.Derives1 \<NN> \<TT> \<RR> a i r b" shows "i < length a"
subnet_oreachable: fixes \<sigma> :: "nat \<Rightarrow> 'd"   and s :: "'e net_state"   and t :: "'e net_state"   and onp :: "nat \<Rightarrow> ((nat \<Rightarrow> 'd) \<times> 'e, 'f seq_action) automaton"   and p\<^sub>1 :: "net_tree"   and p\<^sub>2 :: "net_tree"   and S :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and I :: "(nat \<Rightarrow> 'd) \<Rightarrow> 'f \<Rightarrow> bool"   and U :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "(\<sigma>, SubnetS s t) \<in> oreachable (opnet onp (p\<^sub>1 \<parallel> p\<^sub>2)) (otherwith S (net_tree_ips (p\<^sub>1 \<parallel> p\<^sub>2)) (oarrivemsg I)) (other U (net_tree_ips (p\<^sub>1 \<parallel> p\<^sub>2)))"   and "\<And>\<xi>. S \<xi> \<xi>"   and "\<And>\<xi>. U \<xi> \<xi>"   and "\<And>i R. \<langle>i : onp i : R\<rangle>\<^sub>o \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg I \<sigma>, other U {i} \<rightarrow>) globala (\<lambda>(\<sigma>, a, uu_). castmsg (I \<sigma>) a)"   and "\<And>i R. \<langle>i : onp i : R\<rangle>\<^sub>o \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg I \<sigma>, other U {i} \<rightarrow>) globala (\<lambda>(\<sigma>, a, \<sigma>'). a \<noteq> \<tau> \<and> (\<forall>d. a \<noteq> i:deliver(d)) \<longrightarrow> S (\<sigma> i) (\<sigma>' i))"   and "\<And>i R. \<langle>i : onp i : R\<rangle>\<^sub>o \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg I \<sigma>, other U {i} \<rightarrow>) globala (\<lambda>(\<sigma>, a, \<sigma>'). a = \<tau> \<or> (\<exists>d. a = i:deliver(d)) \<longrightarrow> U (\<sigma> i) (\<sigma>' i))" shows "(\<sigma>, s) \<in> oreachable (opnet onp p\<^sub>1) (otherwith S (net_tree_ips p\<^sub>1) (oarrivemsg I)) (other U (net_tree_ips p\<^sub>1)) \<and> (\<sigma>, t) \<in> oreachable (opnet onp p\<^sub>2) (otherwith S (net_tree_ips p\<^sub>2) (oarrivemsg I)) (other U (net_tree_ips p\<^sub>2)) \<and> net_tree_ips p\<^sub>1 \<inter> net_tree_ips p\<^sub>2 = {}"
DynProcModifyReturnNoAbr: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and init :: "'e \<Rightarrow> 'e"   and p :: "'e \<Rightarrow> 'd"   and return' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and Modif :: "'e \<Rightarrow> 'e set"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P dynCall init p return' c Q,A"   and "\<forall>s t. t \<in> Modif (init s) \<longrightarrow> return' s t = return s t"   and "\<forall>s\<in>P. \<forall>\<sigma>. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/UNIV \<^esub>{\<sigma>} Call (p s) (Modif \<sigma>),{}" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P dynCall init p return c Q,A"