hcju/isabelleps huggingface dataset:
all_security_requirements_fulfilled_mono_stateful_policy_to_network_graph: fixes M :: "'b SecurityInvariant_configured list"   and E' :: "('b \<times> 'b) set"   and E :: "('b \<times> 'b) set"   and V :: "'b set"   and Efix :: "('b \<times> 'b) set" assumes "valid_reqs M"   and "E' \<subseteq> E"   and "wf_graph \<lparr>nodes = V, edges = Efix \<union> E\<rparr>"   and "all_security_requirements_fulfilled M (stateful_policy_to_network_graph \<lparr>hosts = V, flows_fix = Efix, flows_state = E\<rparr>)" shows "all_security_requirements_fulfilled M (stateful_policy_to_network_graph \<lparr>hosts = V, flows_fix = Efix, flows_state = E'\<rparr>)"
eventually_filtercomapI: fixes P :: "'a \<Rightarrow> bool"   and F :: "'a filter"   and f :: "'b \<Rightarrow> 'a" assumes "eventually P F" shows "\<forall>\<^sub>F x in filtercomap f F. P (f x)"
lSup_LCons: fixes A :: "'f llist set"   and x :: "'f" assumes "A \<noteq> {}" shows "lSup (LCons x ` A) = LCons x (lSup A)"
ltl_ldrop: fixes P :: "'c llist \<Rightarrow> bool"   and xs :: "'c llist"   and n :: "nat" assumes "\<And>xs. P xs \<Longrightarrow> P (ltl xs)"   and "P xs" shows "P (ldropn n xs)"
linear_compose: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and f :: "'e \<Rightarrow> 'f"   and s3 :: "'d \<Rightarrow> 'g \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'g" assumes "linear s1 s2 f"   and "linear s2 s3 g" shows "linear s1 s3 (g \<circ> f)"
TensorDiag_Diag: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Diag (t \<^bold>\<otimes> u)" shows "t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u = t \<^bold>\<otimes> u"
is_rendered_commutative_by_cone: fixes a :: "'c"   and \<chi> :: "J.Arr \<Rightarrow> 'c" assumes "D.cone a \<chi>" shows "D.is_rendered_commutative_by (\<chi> J.AA) (\<chi> J.BB)"
check_poly_weak_mono_and_pos_def: fixes discrete :: "bool"   and p :: "('c monom \<times> 'd) list" shows "check_poly_weak_mono_and_pos discrete p \<equiv> if discrete then list_all (check_poly_weak_mono_discrete p) (poly_vars_list p) \<and> (0::'d) \<le> eval_poly (\<lambda>w. 0::'d) p else check_poly_weak_mono_all p"
robot_SPRSingleAlg_def: shows "robot_SPRSingleAlg \<equiv> mkSPRSingleAuto kbp envInit envAction envTrans envVal (\<lambda>_. envObs)"
p_def: fixes s :: "'s option"   and t :: "'s option"   and n :: "nat" shows "K2.p s t n = K2.T.prob s {\<omega> \<in> space (K2.T s). (s ## \<omega>) !! n = t}"
rel_mono: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and Ra :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "R \<le> Ra" shows "rel_pmf R \<le> rel_pmf Ra"
stable_on_Xd: fixes ds :: "'b set"   and X :: "'a set"   and x :: "'a" assumes "Y.stable_on ds X"   and "x \<in> X" shows "Xd x \<in> ds"
Compl_eq_Diff_UNIV: fixes A :: "'d set" shows "- A = UNIV - A"
rel_pmf_return_pmfI: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'b" assumes "P x y" shows "rel_pmf P (return_pmf x) (return_pmf y)"
range_def: fixes index :: "nat \<times> nat" shows "local.range index \<equiv> {..<fst index}"
R: shows "UNlift cl dl P F ` UNpart cl P = UNpart dl (\<lambda>n. F n ` P n)"
path_subpath: fixes g :: "real \<Rightarrow> 'b"   and u :: "real"   and v :: "real" assumes "path g"   and "u \<in> {0..1}"   and "v \<in> {0..1}" shows "path (subpath u v g)"
commutative_monoid_axioms: shows "commutative_monoid A addA zeroA"
inverse_ring_isomorphism: fixes \<eta> :: "'c \<Rightarrow> 'd"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and R' :: "'d set"   and addition' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multiplication' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero' :: "'d"   and unit' :: "'d" assumes "ring_isomorphism \<eta> R addition multiplication zero unit R' addition' multiplication' zero' unit'" shows "ring_isomorphism (restrict (inv_into R \<eta>) R') R' addition' multiplication' zero' unit' R addition multiplication zero unit"
alw_ev_sdrop: fixes \<phi> :: "'b stream \<Rightarrow> bool"   and m :: "nat"   and xs :: "'b stream" assumes "alw (ev \<phi>) (sdrop m xs)" shows "alw (ev \<phi>) xs"
UN_absorb: fixes k :: "'a"   and I :: "'a set"   and A :: "'a \<Rightarrow> 'b set" assumes "k \<in> I" shows "A k \<union> \<Union> (A ` I) = \<Union> (A ` I)"
game_def: fixes b :: "bool"   and \<A> :: "(bool, 'a, 'a) gpv" shows "game b \<A> = run_gpv (if b then rp.random_permutation else rf.random_oracle) \<A> Map.empty"
t2s_def_2: shows "ts' = t2s @ [t]"
IH: assumes "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>s\<^sub>s\<^sub>t A \<cdot>\<^sub>s\<^sub>e\<^sub>t \<delta>)" shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>s\<^sub>s\<^sub>t (A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta>))"
finite_insert: fixes a :: "'c"   and A :: "'c set" shows "finite (Set.insert a A) = finite A"
r_cancel: fixes a :: "'a vec"   and c :: "'a vec"   and b :: "'a vec" assumes "a + c = b + c"   and "a \<in> carrier_vec n"   and "b \<in> carrier_vec n"   and "c \<in> carrier_vec n" shows "a = b"
Eq_Eats_iff: fixes i :: "name"   and z :: "tm"   and t :: "tm"   and u :: "tm"   and H :: "fm set" assumes "atom i \<sharp> (z, t, u)" shows "H \<turnstile> z EQ Eats t u IFF SyntaxN.All i (Var i IN z IFF Var i IN t OR Var i EQ u)"
irreducibleE: fixes G :: "('b, 'c) monoid_scheme"   and a :: "'b" assumes "Divisibility.irreducible G a" obtains "a \<notin> Units G" and "\<forall>b. b \<in> carrier G \<and> Divisibility.properfactor G b a \<longrightarrow> b \<in> Units G"
intro: fixes compile :: "'d \<Rightarrow> 'e option"   and load2 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and load1 :: "'d \<Rightarrow> 'g \<Rightarrow> bool"   and match :: "'h \<Rightarrow> 'g \<Rightarrow> 'f \<Rightarrow> bool" assumes "\<And>p1 p2 s2. \<lbrakk>compile p1 = Some p2; load2 p2 s2\<rbrakk> \<Longrightarrow> \<exists>s1 i. load1 p1 s1 \<and> match i s1 s2" shows "compiler_axioms load1 load2 match compile"
typings_append: fixes CT :: "nat \<Rightarrow> classDef option"   and \<Gamma> :: "nat \<Rightarrow> nat option"   and es :: "exp list"   and Cs :: "nat list"   and e :: "exp"   and C :: "nat" assumes "CT;\<Gamma> \<turnstile>+ es : Cs"   and "CT;\<Gamma> \<turnstile> e : C" shows "CT;\<Gamma> \<turnstile>+ (es @ [e]) : (Cs @ [C])"
lm093: fixes x :: "'d"   and X :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "x \<in> X" shows "toFunction (graph X f) x = f x"
tendsto_lowerbound: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and "open" :: "'c set \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and x :: "'c"   and F :: "'d filter"   and a :: "'c" assumes "class.linorder_topology less_eq less open"   and "filterlim f (topological_space.nhds open x) F"   and "\<forall>\<^sub>F i in F. less_eq a (f i)"   and "F \<noteq> bot" shows "less_eq a x"
split_sorted_1: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and xs :: "('b btree \<times> 'b) list"   and p :: "'b"   and ls :: "('b btree \<times> 'b) list"   and sub :: "'b btree"   and sep :: "'b"   and rs :: "('b btree \<times> 'b) list" assumes "split split"   and "split xs p = (ls @ [(sub, sep)], rs)"   and "sorted_less (separators xs)" shows "sep < p"
\<alpha>r_def: shows "\<alpha>r \<equiv> \<lambda>res v. case res v of None \<Rightarrow> None | Some (p, w) \<Rightarrow> Some (rev p)"
crit_pair_zero: fixes q :: "'t \<Rightarrow>\<^sub>0 'c" shows "fst (crit_pair 0 q) = 0"
known_ptr_impl: shows "known_ptr = ShadowRootClass.known_ptr"
iMODb_append_union_Suc: fixes r :: "nat"   and m :: "nat"   and c :: "nat"   and c' :: "nat" shows "[ r, mod m, c ] \<union> [ r + m * Suc c, mod m, c' ] = [ r, mod m, Suc (c + c') ]"
list_all2_map1: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'a"   and as :: "'c list"   and bs :: "'b list" shows "list_all2 P (map f as) bs = list_all2 (\<lambda>x. P (f x)) as bs"
scaleRat_leq1: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleRat :: "rat \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and k :: "rat" assumes "class.linordered_rational_vector less_eq less plus zero minus uminus scaleRat"   and "less_eq a b"   and "0 < k" shows "less_eq (scaleRat k a) (scaleRat k b)"
lemma_2_10_25: fixes a :: "'a"   and b :: "'a" shows "a \<le> (a l\<rightarrow> b) r\<rightarrow> a"
env_defined: fixes x :: "char list"   and e :: "'b environment" assumes "x \<in> env_dom e" shows "\<exists>T. e!x = Some T"
add_zero_example: shows "(Var 0)\<^sup>2 - Const (1::'a) + Const (0::'a) = (Var 0)\<^sup>2 - Const (1::'a)"
elims_2: fixes x :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xa :: "'a list" assumes "sorted_by x xa" obtains cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where "x = cmp__" and "xa = []"   | cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and uu_ :: "'a" where "x = cmpa__" and "xa = [uu_]"   | cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and x1 :: "'a" and x2 :: "'a" and xs :: "'a list" where "x = cmp" and "xa = x1 # x2 # xs" and "cmp x1 x2 \<and> sorted_by cmp (x2 # xs)"
lossless_resource_of_oracle: fixes callee :: "'h \<Rightarrow> 'i \<Rightarrow> ('j \<times> 'h) spmf"   and I :: "'h \<Rightarrow> bool"   and \<I> :: "('i, 'j) \<I>"   and s :: "'h" assumes "callee_invariant_on callee I \<I>"   and "\<And>s x. \<lbrakk>x \<in> outs_\<I> \<I>; I s\<rbrakk> \<Longrightarrow> lossless_spmf (callee s x)"   and "I s" shows "lossless_resource \<I> (RES callee s)"
cgenideal_minimal: fixes J :: "('c multiset \<Rightarrow> 'a) list set"   and a :: "('c multiset \<Rightarrow> 'a) list" assumes "ideal J (poly_ring L)"   and "a \<in> J" shows "PIdl\<^bsub>poly_ring L\<^esub> a \<subseteq> J"
y: shows "y = root 3 u - e / (3 * root 3 u)"
section_retraction_of_iso_2: fixes f :: "'a"   and g :: "'a" assumes "local.iso f"   and "ide (f \<cdot> g)" shows "inverse_arrows f g"
intros_11: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and chain :: "char list"   and rs :: "'c rule list"   and t :: "state" assumes "matches \<gamma> m p"   and "\<Gamma> chain = Some rs"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> t" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m (Call chain)], Undecided\<rangle> \<Rightarrow> t"
not_sym: fixes t :: "'e"   and s :: "'e" assumes "t \<noteq> s" shows "s \<noteq> t"
nonzero_divide_mult_cancel_left: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> (0::'a)" shows "a / (a * b) = (1::'a) / b"
assms_2: shows "g piecewise_differentiable_on {0..1}"
stone_phi_embed_order_injective: fixes x :: "'c stone_phi_pair"   and y :: "'c stone_phi_pair" assumes "stone_phi_embed x \<le> stone_phi_embed y" shows "x \<le> y"
SC_sublist2: fixes xs :: "('b, 'c) Combinators list"   and P :: "('b, 'c) Combinators \<Rightarrow> bool" assumes "singleCombinators xs" shows "singleCombinators (filter P xs)"
tab_vn: shows "(tab(x # xs [\<mapsto>] ys)) vn = Some el"
map\<^sub>0_simps_3: fixes a :: "'a" assumes "C.obj a" shows "trg\<^sub>D (G.map\<^sub>0 a) = G.map\<^sub>0 a"
g: fixes t :: "real" assumes "0 \<le> t"   and "t \<le> 1" shows "g t = c"
AbGroup_inner_dirsum_el_decomp_nth_id_on_nth: fixes Gs :: "'a set list"   and n :: "nat"   and x :: "'a" assumes "\<forall>G\<in>set Gs. AbGroup G"   and "add_independentS Gs"   and "n < length Gs"   and "x \<in> Gs ! n" shows "(if x \<in> (\<Oplus>G\<leftarrow>Gs. G) then \<Oplus>Gs\<leftarrow> x ! n else (0::'a)) = x"
2: shows "Fvars \<chi> - set us = Fvars \<phi> - snd ` set txs"
that: fixes \<Gamma>' :: "'a formula list" assumes "set \<Gamma>' \<subseteq> \<Gamma> \<and> ({} \<turnstile> \<^bold>\<And>\<Gamma>' \<^bold>\<rightarrow> F)" shows "thesis"
continuous_map_openin_preimage_eq: fixes X :: "'d topology"   and Y :: "'e topology"   and f :: "'d \<Rightarrow> 'e" shows "continuous_map X Y f = (f ` topspace X \<subseteq> topspace Y \<and> (\<forall>U. openin Y U \<longrightarrow> openin X (topspace X \<inter> f -` U)))"
indep_vars_cong: fixes I :: "'c set"   and J :: "'c set"   and X :: "'c \<Rightarrow> 'a \<Rightarrow> 'd"   and Y :: "'c \<Rightarrow> 'a \<Rightarrow> 'd"   and M' :: "'c \<Rightarrow> 'd measure"   and N' :: "'c \<Rightarrow> 'd measure" assumes "I = J"   and "\<And>i. i \<in> I \<Longrightarrow> X i = Y i"   and "\<And>i. i \<in> I \<Longrightarrow> M' i = N' i" shows "indep_vars M' X I = indep_vars N' Y J"
atoms_flatten_PLUS: fixes X :: "'b rexp set" assumes "finite X" shows "atoms (flatten PLUS X) = \<Union> (atoms ` X)"
subst_trm_ty_def: shows "subst_trm_ty \<equiv> \<lambda>x Y T2. trm_rec Var (\<lambda>x t T. Abs_syn x (T[Y \<mapsto> T2]\<^sub>\<tau>)) (\<lambda>X t T. TAbs_syn X (T[Y \<mapsto> T2]\<^sub>\<tau>)) (\<lambda>t1 t2. (\<cdot>)) (\<lambda>t1 T t1a. t1a \<cdot>\<^sub>\<tau> T[Y \<mapsto> T2]\<^sub>\<tau>) x"
representation_def: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and basis :: "'d set"   and v :: "'d" assumes "module scale" shows "module.representation scale basis v = (if \<not> module.dependent scale basis \<and> v \<in> module.span scale basis then SOME f. (\<forall>v. f v \<noteq> (0::'c) \<longrightarrow> v \<in> basis) \<and> finite {v. f v \<noteq> (0::'c)} \<and> (\<Sum>v | f v \<noteq> (0::'c). scale (f v) v) = v else (\<lambda>b. 0::'c))"
simps: fixes a1 :: "('procS, 'procT) Proc"   and a2 :: "('procS, 'procT) Proc" shows "a1 \<lesssim>\<lbrakk>\<cdot>\<rbrakk>R a2 = ((\<exists>S. S \<in>S a1 \<and> \<lbrakk>S\<rbrakk> \<in>T a2) \<or> (\<exists>S. S \<in>S a1 \<and> S \<in>S a2) \<or> (\<exists>T. T \<in>T a1 \<and> T \<in>T a2) \<or> (\<exists>P Q R. a1 = P \<and> a2 = R \<and> P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>R Q \<and> Q \<lesssim>\<lbrakk>\<cdot>\<rbrakk>R R))"
l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and first_in_tree_order :: "(_) object_ptr \<Rightarrow> ((_) object_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog) \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and get_attribute :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, char list option) prog"   and get_attribute_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_element_by_id :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and get_elements_by_class_name :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr list) prog"   and get_elements_by_tag_name :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr list) prog"   and type_wf :: "(_) heap \<Rightarrow> bool" shows "l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M to_tree_order first_in_tree_order get_attribute get_attribute_locs get_element_by_id get_elements_by_class_name get_elements_by_tag_name type_wf \<equiv> (l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M to_tree_order first_in_tree_order \<and> l_to_tree_order to_tree_order) \<and> l_get_attribute type_wf get_attribute get_attribute_locs \<and> l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms to_tree_order first_in_tree_order get_attribute get_element_by_id get_elements_by_class_name get_elements_by_tag_name"
rel_I2: fixes s :: "'b"   and U :: "'a set"   and x :: "'a set \<times> 'b" assumes "s \<in> \<FF> U"   and "x \<in> \<lfloor>U, s\<rfloor>"   and "U \<in> I" shows "(U, s) \<sim> (SOME x. x \<in> \<lfloor>U, s\<rfloor>)"
positively_invariant_outer: shows "c.positively_invariant {p. (fst p)\<^sup>2 + (snd p)\<^sup>2 - 4 \<le> 0}"
e_b_c_a: fixes b :: "'a"   and c :: "'a"   and a :: "'a" shows "e_aux b c a = e_aux a b c"
lele: shows "(q < x in A \<longrightarrow> q < x in swaps [index A q - n..<index A q] A) \<and> index (swaps [index A q - n..<index A q] A) q = index A q - n"
is_tm_dghm_axioms: shows "\<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>"
isecure_total_def: shows "int.isecure_total \<equiv> int.NI_unrelated_total \<and> int.NI_indirect_sources_total"
wp_Seq: fixes P :: "'b \<Rightarrow> real"   and a :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and Q :: "'b \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and R :: "'b \<Rightarrow> real" assumes "P \<tturnstile> wp a Q"   and "Q \<tturnstile> wp b R"   and "well_def a"   and "well_def b"   and "sound Q"   and "sound R" shows "P \<tturnstile> wp (a ;; b) R"
unitary11_gen_mult_sm: fixes k :: "complex"   and M :: "complex \<times> complex \<times> complex \<times> complex" assumes "k \<noteq> 0"   and "unitary11_gen M" shows "unitary11_gen (k *\<^sub>s\<^sub>m M)"
ee: fixes z :: "complex" assumes "z \<in> path_image p" shows "0 < ee z \<and> ball z (ee z) \<subseteq> S"
subst_clss_image: fixes f :: "'b \<Rightarrow> 'a literal multiset"   and X :: "'b set"   and \<sigma> :: "'s" shows "f ` X \<cdot>cs \<sigma> = {f x \<cdot> \<sigma> |x. x \<in> X}"
ExpList_rep: fixes z :: "exp list" shows "\<exists>Us. z = Abs_ExpList Us"
mono: fixes x :: "('b, 'call, 'ret) gpv \<times> 's" shows "spmf.mono_body (\<lambda>inline1'. case x of (gpv, s) \<Rightarrow> the_gpv gpv \<bind> case_generat (\<lambda>x. return_spmf (Inl (x, s))) (\<lambda>out rpv. the_gpv (callee s out) \<bind> case_generat (\<lambda>(x, y). curry inline1' (rpv x) y) (\<lambda>out' rpv'. return_spmf (Inr (out, out', rpv', rpv)))))"
assms: shows "x \<cdot> y = y \<cdot> x"
smult_1_left: fixes p :: "'c poly" shows "smult (1::'c) p = p"
rref_condition3_equiv: fixes A :: "(('a, 'b) vec, 'c) vec"   and i :: "'c"   and j :: "'c" assumes "reduced_row_echelon_form A"   and "i < j"   and "\<not> is_zero_row i A"   and "\<not> is_zero_row j A" shows "(LEAST n. A $ i $ n \<noteq> (0::'a)) < (LEAST n. A $ j $ n \<noteq> (0::'a))"
image_lepoll: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "f ` A \<lesssim> A"
sel_1: fixes x11 :: "'a"   and x12 :: "'a process" shows "prefOf (Action x11 x12) = x11"
jkbpC_represents_uniquely: fixes T :: "'s Trace set" assumes "Clock.represents T" shows "T = Clock.jkbpC"
tllist_all2_tmap2: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "('a, 'c) tllist"   and f :: "'e \<Rightarrow> 'b"   and g :: "'f \<Rightarrow> 'd"   and ys :: "('e, 'f) tllist" shows "tllist_all2 P Q xs (tmap f g ys) = tllist_all2 (\<lambda>x y. P x (f y)) (\<lambda>x y. Q x (g y)) xs ys"
lookup_map: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and t :: "('b, 'c) RBT.rbt"   and k :: "'b" shows "RBT.lookup (RBT.map f t) k = map_option (f k) (RBT.lookup t k)"
const_def: fixes n :: "nat" shows "const n = (\<forall>x\<in>{0, 1}. f x = n)"
collapse: fixes prod :: "'g \<times> 'h" shows "(fst prod, snd prod) = prod"
fold_const: fixes xs :: "'d list"   and a :: "'d" shows "fold (\<lambda>x _. x) xs a = last (a # xs)"
sgn_blinfun_def: fixes x :: "'a \<Rightarrow>\<^sub>L 'b" shows "sgn x = inverse (norm x) *\<^sub>R x"
hom_add: fixes x :: "'b poly"   and y :: "'b poly"   and a :: "'b" shows "poly (x + y) a = poly x a + poly y a"
sum_all0_intro: fixes b :: "'a"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "b = (0::'a)"   and "\<And>a. a \<in> A \<Longrightarrow> f a = (0::'a)" shows "sum f A = b"
redundancy_criterion_axioms: shows "redundancy_criterion gr.ord_\<Gamma> sr.Rf sr.Ri"
strand_sem_append_2: fixes M :: "('fun, 'var) Term.term set"   and S :: "('fun, 'var) strand_step list"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S' :: "('fun, 'var) strand_step list" assumes "\<lbrakk>M; S\<rbrakk>\<^sub>d \<theta>"   and "\<lbrakk>M \<union> (ik\<^sub>s\<^sub>t S \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>); S'\<rbrakk>\<^sub>d \<theta>" shows "\<lbrakk>M; S @ S'\<rbrakk>\<^sub>d \<theta>"
LiberalConseq': fixes P :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and c :: "('d, 'e, 'f) com"   and Q :: "'d set"   and A :: "'d set" assumes "\<forall>s\<in>P. \<exists>P' Q' A'. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P' c Q',A' \<and> (\<forall>t. (s \<in> P' \<longrightarrow> t \<in> Normal ` Q' \<union> Abrupt ` A') \<longrightarrow> t \<in> Normal ` Q \<union> Abrupt ` A)" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
prj_simps_11: shows "C.seq doms.\<nu>\<pi>.prj\<^sub>1 \<p>\<^sub>0[doms.\<mu>.leg0, doms.\<nu>.leg1 \<cdot> doms.\<nu>\<pi>.prj\<^sub>1]"
SystemDWSubst_def: shows "SystemDWSubst = \<lparr>SFunctions = Map.empty, SPredicates = Map.empty, SContexts = \<lambda>C. if C = pid1 then Some (Geq (Const 0) (Const 0) && Geq ($f fid1 local.empty) (Const 0)) else None, SPrograms = Map.empty, SODEs = \<lambda>c. if c = vid1 then Some (OProd (OSing vid1 ($f fid1 local.empty)) (OSing vid2 (Var vid1))) else None\<rparr>"
bound: fixes P :: "'a \<Rightarrow> ennreal" shows "f P \<le> f (\<lambda>s. emeasure (M s) (space (M s)))"
assms_1: shows "f \<in> borel_measurable lebesgue"
mset_add: fixes a :: "'b"   and A :: "'b multiset" assumes "a \<in># A" obtains B :: "'b multiset" where "A = a, B"
up_defined: fixes x :: "'c" shows "up\<cdot>x \<noteq> \<bottom>"
is_ring_morphism: fixes U :: "'d set"   and V :: "'d set" assumes "U = {}"   and "V = {}"   and "V \<subseteq> U" shows "ring_homomorphism (im0.im_sheaf_morphisms U V) (im0.im_sheaf U) (im0.add_im_sheaf U) (im0.mult_im_sheaf U) (im0.zero_im_sheaf U) (im0.one_im_sheaf U) (im0.im_sheaf V) (im0.add_im_sheaf V) (im0.mult_im_sheaf V) (im0.zero_im_sheaf V) (im0.one_im_sheaf V)"
update_unsat_core_id: fixes update :: "nat \<Rightarrow> 'a \<Rightarrow> ('b, 'a) state \<Rightarrow> ('b, 'a) state"   and s :: "('b, 'a) state"   and x :: "nat"   and c :: "'a" assumes "Update update"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x \<notin> lvars (\<T> s)" shows "\<U>\<^sub>c (update x c s) = \<U>\<^sub>c s"
bind_spmf_assoc: fixes x :: "'f spmf"   and f :: "'f \<Rightarrow> 'e spmf"   and g :: "'e \<Rightarrow> 'd spmf" shows "x \<bind> f \<bind> g = x \<bind> (\<lambda>y. f y \<bind> g)"
Is: shows "\<forall>s1 s2 l. s1 = s2 on Is \<longrightarrow> I l s1 = I l s2"
PHI: fixes s :: "'b" assumes "I s"   and "\<not> b s" shows "\<Phi> s"
squash_pow: fixes x :: "real"   and y :: "nat" shows "squash (x ^ y) = squash x ^ y"
rbt_less_prop: fixes k :: "'a"   and t :: "('a, 'd) rbt" shows "(t |\<guillemotleft> k) = (\<forall>x\<in>set (keys t). x < k)"
fold_is_None: fixes x :: "'b option" shows "(x = None) = is_None x"
eliminate_entries_gen_zero: fixes A :: "'i mat"   and A' :: "'a mat"   and I :: "nat"   and v :: "integer \<Rightarrow> 'i"   and J :: "nat" assumes "mat_rel R A A'"   and "I < dim_row A'" shows "eliminate_entries_gen_zero (arith_ops_record.minus ops) (arith_ops_record.times ops) (arith_ops_record.zero ops) v A I J = eliminate_entries_gen (arith_ops_record.minus ops) (arith_ops_record.times ops) (v \<circ> integer_of_nat) A I J"
in_progressS_med_progress: fixes r :: "fid_t \<Rightarrow> Runs.var set option"   and R :: "fid_t"   and S :: "Runs.var set" assumes "in_progressS (r R) S" shows "in_progressS (med_progress r R) (S \<inter> {xnx, xny, xgnx, xgny, xsk, xEnd})"
subset_dual: fixes X :: "'a set"   and Y :: "'a set" shows "(X = \<partial> ` Y) = (\<partial> ` X = Y)"
P2_xor_inf_th: fixes A :: "bool \<times> bool"   and B :: "bool \<times> bool" shows "R2_xor A B = xor_funct A B \<bind> (\<lambda>(out1, out2). S2_xor B out2)"