hcju/isabelleps huggingface dataset:
exec_impl_steps: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and t :: "('b, 'c) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t" shows "\<exists>c' t'. \<Gamma>\<turnstile> (c, s) \<rightarrow>\<^sup>* (c', t') \<and> (case t of Abrupt x \<Rightarrow> if s = t then c' = Skip \<and> t' = t else c' = Throw \<and> t' = Normal x | _ \<Rightarrow> c' = Skip \<and> t' = t)"
mat_of_row_dim_2: fixes y :: "'a vec" shows "dim_col (mat_of_row y) = dim_vec y"
node_disconnectTE': fixes ii :: "nat"   and s :: "'c"   and R :: "nat set"   and i :: "nat"   and i' :: "nat"   and s' :: "'c net_state"   and S :: "('c \<times> 'd seq_action \<times> 'c) set" assumes "(NodeS ii s R, disconnect(i, i'), s') \<in> node_sos S" obtains "i = ii" and "s' = NodeS ii s (R - {i'})"   | "i' = ii" and "s' = NodeS ii s (R - {i})"   | "s' = NodeS ii s R" and "ii \<noteq> i" and "ii \<noteq> i'"
map_def: fixes f :: "'j" shows "A'.map f \<equiv> if J.arr f then a' else C.null"
le_infI2: fixes b :: "'b multiset"   and x :: "'b multiset"   and a :: "'b multiset" assumes "b \<subseteq># x" shows "a \<inter># b \<subseteq># x"
induct: fixes x :: "('a, 'b) form list"   and P :: "('a, 'b) form list \<Rightarrow> bool" assumes "\<stileturn> x"   and "\<And>i l G. P (Pred i l # Neg (Pred i l) # G)"   and "\<And>G. P (\<bottom> # G)"   and "\<And>G. P (Neg \<top> # G)"   and "\<And>A G. \<lbrakk>\<stileturn> A # G; P (A # G)\<rbrakk> \<Longrightarrow> P (Neg (Neg A) # G)"   and "\<And>A B G. \<lbrakk>\<stileturn> A # B # G; P (A # B # G)\<rbrakk> \<Longrightarrow> P (And A B # G)"   and "\<And>A B G. \<lbrakk>\<stileturn> Neg A # Neg B # G; P (Neg A # Neg B # G)\<rbrakk> \<Longrightarrow> P (Neg (Or A B) # G)"   and "\<And>A B G. \<lbrakk>\<stileturn> A # Neg B # G; P (A # Neg B # G)\<rbrakk> \<Longrightarrow> P (Neg (Impl A B) # G)"   and "\<And>A G B. \<lbrakk>\<stileturn> Neg A # G; P (Neg A # G); \<stileturn> Neg B # G; P (Neg B # G)\<rbrakk> \<Longrightarrow> P (Neg (And A B) # G)"   and "\<And>A G B. \<lbrakk>\<stileturn> A # G; P (A # G); \<stileturn> B # G; P (B # G)\<rbrakk> \<Longrightarrow> P (Or A B # G)"   and "\<And>A G B. \<lbrakk>\<stileturn> Neg A # G; P (Neg A # G); \<stileturn> B # G; P (B # G)\<rbrakk> \<Longrightarrow> P (Impl A B # G)"   and "\<And>A t G. \<lbrakk>\<stileturn> A[t/0] # G; P (A[t/0] # G)\<rbrakk> \<Longrightarrow> P (Forall A # G)"   and "\<And>A t G. \<lbrakk>\<stileturn> Neg (A[t/0]) # G; P (Neg (A[t/0]) # G)\<rbrakk> \<Longrightarrow> P (Neg (Exists A) # G)"   and "\<And>A n G. \<lbrakk>\<stileturn> A[App n []/0] # G; P (A[App n []/0] # G); news n (A # G)\<rbrakk> \<Longrightarrow> P (Exists A # G)"   and "\<And>A n G. \<lbrakk>\<stileturn> Neg (A[App n []/0]) # G; P (Neg (A[App n []/0]) # G); news n (A # G)\<rbrakk> \<Longrightarrow> P (Neg (Forall A) # G)"   and "\<And>G G'. \<lbrakk>\<stileturn> G; P G; set G = set G'\<rbrakk> \<Longrightarrow> P G'" shows "P x"
that: fixes A' :: "'a set" assumes "A \<subseteq> A'"   and "A' \<in> sets M"   and "A' - A \<in> null_sets (completion M)"   and "emeasure (completion M) A = emeasure M A'" shows "thesis"
trace_preserves_read_only_unowned: fixes c :: "nat \<Rightarrow> ('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option)"   and n :: "nat"   and k :: "nat"   and l :: "nat" assumes "simple_ownership_distinct (fst (c n))"   and "read_only_unowned (snd (snd (c n))) (fst (c n))"   and "trace c n k"   and "\<forall>x<k. safe_delayed (c (n + x))"   and "l \<le> k" shows "read_only_unowned (snd (snd (c (n + l)))) (fst (c (n + l)))"
dg_singleton_ObjI: fixes A :: "V"   and j :: "V"   and a :: "V"   and \<CC> :: "V" assumes "A = ZFC_in_HOL.set {\<langle>j, a\<rangle>}"   and "a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>" shows "A \<in>\<^sub>\<circ> (\<Prod>\<^sub>D\<^sub>Gi\<in>\<^sub>\<circ>ZFC_in_HOL.set {j}. \<CC>)\<lparr>Obj\<rparr>"
sscan_sdrop: fixes k :: "nat"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and xs :: "'d stream"   and a :: "'c" shows "sdrop k (sscan f xs a) = sscan f (sdrop k xs) (fold f (stake k xs) a)"
insert_not_empty: fixes a :: "'b"   and A :: "'b set" shows "(a::'b) \<triangleright> (A::'b set) \<noteq> {}"
\<omega>_sum_append: fixes xs :: "V list"   and ys :: "V list" shows "\<omega>_sum (xs @ ys) = \<omega>_sum xs + \<omega>_sum ys"
gap_pres_hb_labeling: fixes k :: "nat" assumes "gap_precond l k" shows "Height_Bounded_Labeling c s t f (gap_effect l k)"
xs_1: shows "xs \<longlonglongrightarrow> x"
collapse: fixes tree :: "'b tree" assumes "tree \<noteq> Leaf" shows "Node (l tree) (r tree) (val tree) (lchild tree) (rchild tree) = tree"
diff_strict_left_mono: fixes b :: "complex"   and a :: "complex"   and c :: "complex" assumes "b <\<^sub>\<complex> a" shows "c - a <\<^sub>\<complex> c - b"
poly_mult_rat_def: fixes r :: "rat"   and p :: "int poly" shows "poly_mult_rat r p \<equiv> case quotient_of r of (n, d) \<Rightarrow> poly_mult_rat_main n d p"
abs_eq: fixes x :: "int" shows "uint8_of_int x = Abs_uint8 (word_of_int x)"
grid_ge_min: fixes j :: "nat" shows "t 0 \<le> t j"
asms_1: shows "list_all (\<lambda>x. x < Suc i) xs"
apx_V: fixes Z :: "('c \<Rightarrow> real) set" assumes "Z \<subseteq> V" shows "Approx\<^sub>\<beta> Z \<subseteq> V"
base_6: shows "\<forall>i\<in>R. \<forall>j\<in>R. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
prefix_eq_fin_transp: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item list"   and w\<^sub>3 :: "'item list" assumes "w\<^sub>1 \<le> w\<^sub>2"   and "w\<^sub>2 =\<^sub>F w\<^sub>3" shows "w\<^sub>1 \<preceq>\<^sub>F w\<^sub>3"
finite_lepoll_infinite: fixes A :: "'a set"   and B :: "'b set" assumes "infinite A"   and "finite B" shows "B \<lesssim> A"
simps_3: fixes n :: "nat"   and v :: "nat"   and t :: "nat ifex"   and e :: "nat ifex"   and bdd :: "bdd" shows "Rmi_g (Suc (Suc n)) (IF v t e) bdd = (pointermap_p_valid n (dpm bdd) \<and> (case pm_pth (dpm bdd) n of (nv, nt, ne) \<Rightarrow> nv = v \<and> Rmi_g nt t bdd \<and> Rmi_g ne e bdd))"
iffb_ifbd_dual: fixes \<phi> :: "'c set \<Rightarrow> 'd set" assumes "Inf_pres \<phi>" shows "fb\<^sup>-\<^sub>\<F> \<phi> = (op\<^sub>K \<circ> bd\<^sup>-\<^sub>\<F> \<circ> \<partial>\<^sub>F) \<phi>"
meet_cong_l: fixes x :: "'a"   and x' :: "'a"   and y :: "'a" assumes "x \<in> carrier (division_rel G)"   and "x' \<in> carrier (division_rel G)"   and "y \<in> carrier (division_rel G)"   and "x .=\<^bsub>division_rel G\<^esub> x'" shows "x \<sqinter>\<^bsub>division_rel G\<^esub> y .=\<^bsub>division_rel G\<^esub> x' \<sqinter>\<^bsub>division_rel G\<^esub> y"
continuous_on_path: fixes f :: "real \<Rightarrow> 'b"   and t :: "real set" assumes "path f"   and "t \<subseteq> {0..1}" shows "continuous_on t f"
resl_antitoner: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<le> y \<longrightarrow> z \<leftarrow> y \<le> z \<leftarrow> x"
case_map_resumption: fixes "done" :: "'e option \<Rightarrow> 'd"   and pause :: "'f \<Rightarrow> ('g \<Rightarrow> ('e, 'f, 'g) Resumption.resumption) \<Rightarrow> 'd"   and f :: "'h \<Rightarrow> 'e"   and g :: "'i \<Rightarrow> 'f"   and r :: "('h, 'i, 'g) Resumption.resumption" shows "(case Resumption.resumption.map_resumption f g r of Resumption.resumption.Done x \<Rightarrow> done x | Resumption.resumption.Pause x xa \<Rightarrow> pause x xa) = (case r of Resumption.resumption.Done x \<Rightarrow> (done \<circ> map_option f) x | Resumption.resumption.Pause out c \<Rightarrow> pause (g out) (Resumption.resumption.map_resumption f g \<circ> c))"
simps: fixes k :: "nat"   and xs :: "'a list" shows "fast_select k xs = (if length xs \<le> 20 then sort xs ! k else let x = fast_select (((length xs + 4) div 5 - 1) div 2) (map median_le_5 (chop 5 xs)); (ls, es, gs) = threeway_partition x xs in if k < length ls then fast_select k ls else if k < length ls + length es then x else fast_select (k - length ls - length es) gs)"
prime_factorization_prod: 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 :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> zero" shows "factorial_semiring.prime_factorization zero times one normalize (comm_monoid_mult.prod times one f A) = (\<Sum>n\<in>A. factorial_semiring.prime_factorization zero times one normalize (f n))"
segment_inc1: fixes D :: "('c, 'd) Order_scheme"   and b :: "'c"   and a :: "'c" assumes "Order D"   and "b \<in> carrier D" shows "(a \<prec>\<^bsub>D\<^esub> b \<and> a \<in> carrier D) = (a \<in> segment D b)"
disjoint: fixes A :: "'a set"   and B :: "'a set" assumes "A \<in> left.transformation.orbit.Partition"   and "B \<in> left.transformation.orbit.Partition"   and "A \<noteq> B" shows "A \<inter> B = {}"
map_of_clearjunk: fixes al :: "('d \<times> 'e) list" shows "map_of (clearjunk al) = map_of al"
subset_is_non_overlapping: fixes P :: "'c set set"   and Q :: "'c set set" assumes "P \<subseteq> Q"   and "is_non_overlapping Q" shows "is_non_overlapping P"
simps: fixes x :: "'g"   and xs :: "'g list" shows "last (x # xs) = (if xs = [] then x else last xs)"
generalized_fw_joinI: fixes f1 :: "('a simple_match \<times> 'b) list"   and p :: "('a, 'c) simple_packet_scheme"   and r1 :: "'a simple_match"   and d1 :: "'b"   and f2 :: "('a simple_match \<times> 'd) list"   and r2 :: "'a simple_match"   and d2 :: "'d" assumes "generalized_sfw f1 p = Some (r1, d1)"   and "generalized_sfw f2 p = Some (r2, d2)" shows "generalized_sfw (generalized_fw_join f1 f2) p = Some (the (simple_match_and r1 r2), d1, d2)"
k_2: shows "\<forall>x. k (1, x) = h x"
orthogonal_complement_def: fixes W :: "'c vec set" shows "orthogonal_complement W = {x \<in> carrier_vec n. \<forall>y\<in>W. x \<bullet> y = (0::'c)}"
to_gbg_alt1_2: fixes m :: "'b" shows "g_V0 (to_gbg_ext m) = V0"
assms_5: shows "\<forall>m. is_zero_row_upt_k m k A"
alloc_new_instance_in_Store: fixes C :: "CTypeId"   and s :: "Store" shows "Rep_Store s\<lparr>newOID := \<lambda>D. if C = D then Suc (newOID (Rep_Store s) C) else newOID (Rep_Store s) D\<rparr> \<in> Store"
not_outside: fixes x :: "'a"   and y :: "'a" assumes "x \<succeq>[relation] y" shows "x \<in> carrier"
not_arr_null: shows "\<not> H.VV.arr H.VV.null"
inj_on_insert: fixes f :: "'b \<Rightarrow> 'c"   and a :: "'b"   and A :: "'b set" shows "inj_on f (a \<triangleright> A) = (inj_on f A \<and> f a \<notin> f ` (A - {a}))"
weakComm1: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>Q :: "'b"   and R :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>R :: "name list"   and \<Psi>\<^sub>R :: "'b"   and Q :: "('a, 'b, 'c) psi"   and K :: "'a"   and xvec :: "name list"   and Q' :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q : R \<rhd> P \<Longrightarrow>M\<lparr>N\<rparr> \<prec> P'"   and "extractFrame R = \<langle>A\<^sub>R, \<Psi>\<^sub>R\<rangle>"   and "\<Psi> \<otimes> \<Psi>\<^sub>R \<rhd> Q \<longmapsto> K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "\<Psi> \<otimes> \<Psi>\<^sub>R \<otimes> \<Psi>\<^sub>Q \<turnstile> M \<leftrightarrow> K"   and "A\<^sub>R \<sharp>* \<Psi>"   and "A\<^sub>R \<sharp>* P"   and "A\<^sub>R \<sharp>* Q"   and "A\<^sub>R \<sharp>* R"   and "A\<^sub>R \<sharp>* M"   and "A\<^sub>R \<sharp>* A\<^sub>Q"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* R"   and "A\<^sub>Q \<sharp>* K"   and "xvec \<sharp>* P" shows "\<Psi> \<rhd> P \<parallel> Q \<Longrightarrow>\<^sub>\<tau> \<lparr>\<nu>*xvec\<rparr>P' \<parallel> Q'"
\<phi>\<psi>: shows "meta_adjunction (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) F G \<phi> \<psi>"
trace_eq_obsf_resourceI: fixes A :: "'y set"   and res1 :: "('y, 'z) resource"   and res2 :: "('y, 'z) resource" assumes "A \<turnstile>\<^sub>R res1 \<approx> res2" shows "A \<turnstile>\<^sub>R obsf_resource res1 \<approx> obsf_resource res2"
complement: fixes a :: "'b set"   and A :: "'b set set" assumes "a \<in> \<sigma>_algebra A" shows "- a \<in> \<sigma>_algebra A"
gderiv_def: fixes f :: "'a \<Rightarrow> real"   and x :: "'a"   and D :: "'a" shows "GDERIV f x :> D = (f has_derivative (\<lambda>h. inner h D)) (at x)"
UP_ring_subring_1: fixes S :: "'a set" assumes "subring S R" shows "UP_cring (R\<lparr>carrier := S\<rparr>)"
is_weakly_scdom_component_safe_def: shows "Core_DOM_SC_DOM_Components.is_weakly_scdom_component_safe \<equiv> l_get_scdom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_is_weakly_scdom_component_safe Core_DOM_Functions.get_owner_document get_disconnected_nodes Core_DOM_Functions.to_tree_order"
is_processT8_Pair: fixes s :: "'c event list \<times> 'c event set"   and P :: "'c process" assumes "fst s \<in> D P" shows "s \<in> F P"
inv_ide: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "category C"   and "partial_magma.ide C a" shows "category.inv C a = a"
x_img: shows "\<sigma> x \<notin> subst_range \<sigma>"
mod_mult_self2_is_0: 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 a :: "'a"   and b :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (times a b) b = zero"
convergent_cong: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "\<forall>\<^sub>F x in sequentially. f x = g x" shows "convergent f = convergent g"
submultiplicative_converges: fixes u :: "nat \<Rightarrow> real" assumes "\<And>n. 0 \<le> u n"   and "\<And>m n. u (m + n) \<le> u m * u n" shows "(\<lambda>n. root n (u n)) \<longlonglongrightarrow> Inf {root n (u n) |n. 0 < n}"
orderE: fixes a :: "'k multiset"   and b :: "'k multiset" assumes "a \<subseteq># b" obtains "a = a \<inter># b"
list_all2_same: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list" shows "list_all2 P xs xs = (\<forall>x\<in>set xs. P x x)"
dom_char: fixes f :: "('a, 'a set) Maps.arr" shows "Maps.dom f = (if Maps.arr f then Maps.MkIde (Maps.Dom f) else Maps.null)"
distinct_37: fixes psiCase :: "('d, 'e, 'f) psiCase"   and psi1' :: "('d, 'e, 'f) psi"   and psi2' :: "('d, 'e, 'f) psi" shows "Case psiCase \<noteq> psi1' \<parallel> psi2'"
assms_1: fixes dip :: "nat" assumes "dip \<in> kD (rt \<xi>)" shows "dip \<in> kD (rt \<xi>')"
thm_cont_propos_3: fixes v :: "i"   and p :: "\<o>" shows "[Contingent p \<^bold>\<equiv> Contingent (p\<^sup>-) in v]"
F'_def: shows "F' \<equiv> \<lambda>M i. if integrable M f then integral\<^sup>L M (F i) else (0::'b)"
mono'': shows "mono (\<lambda>f. U (F (C f)))"
results_callee: fixes x :: "'call"   and s :: "'s" assumes "x \<in> outs_\<I> \<I>"   and "I s" shows "results_gpv \<I>' (callee s x) \<subseteq> responses_\<I> \<I> x \<times> {s. I s}"
inj_idom_hom_def: fixes hom :: "'b \<Rightarrow> 'c" shows "inj_idom_hom hom \<equiv> idom_hom hom \<and> zero_hom_0 hom"
Diff_subset: fixes A :: "'a set"   and B :: "'a set" shows "A - B \<subseteq> A"
division: fixes d :: "'a set set"   and a :: "'a"   and b :: "'a" assumes "d division_of cbox a b" shows "F (\<lambda>i. if g integrable_on i then Some (integral i g) else None) d = (if g integrable_on cbox a b then Some (integral (cbox a b) g) else None)"
Amax_memTr: fixes f :: "nat \<Rightarrow> ant"   and n :: "nat" shows "f \<in> {i. i \<le> n} \<rightarrow> Z\<^sub>-\<^sub>\<infinity> \<longrightarrow> Amax n f \<in> Z\<^sub>-\<^sub>\<infinity>"
map_spmf_conv_bind_spmf: fixes f :: "'d \<Rightarrow> 'c"   and p :: "'d spmf" shows "map_spmf f p = p \<bind> (\<lambda>x. return_spmf (f x))"
lfp_mono: fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" assumes "mono_mono F" shows "mono (lfp F)"
cauchyX: fixes \<epsilon> :: "real"   and n :: "nat"   and m :: "nat" assumes "0 < \<epsilon>"   and "I \<epsilon> \<le> n"   and "I \<epsilon> \<le> m" shows "norm (X n - X m) \<le> \<epsilon>"
inj_Inl: fixes A :: "'c set" shows "inj_on Inl A"
bij_betw_byWitness: fixes A :: "'a set"   and f' :: "'b \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'b"   and A' :: "'b set" assumes "\<forall>a\<in>A. f' (f a) = a"   and "\<forall>a'\<in>A'. f (f' a') = a'"   and "f ` A \<subseteq> A'"   and "f' ` A' \<subseteq> A" shows "bij_betw f A A'"
ceiling_le_zero: fixes x :: "'b" shows "(\<lceil>x\<rceil> \<le> 0) = (x \<le> (0::'b))"
check_no_overlap2_def: fixes \<gamma> :: "'d set \<Rightarrow> 'e \<Rightarrow> bool"   and ft :: "('d, 'f) flow_entry_match list" shows "check_no_overlap2 \<gamma> ft = (\<forall>a\<in>set ft. \<forall>b\<in>set ft. a \<noteq> b \<and> ofe_prio a = ofe_prio b \<longrightarrow> \<not> (\<exists>p\<in>UNIV. \<gamma> (ofe_fields a) p \<and> \<gamma> (ofe_fields b) p))"
run_state_bind: fixes x :: "('f, 'g) state"   and f :: "'g \<Rightarrow> ('f, 'e) state"   and s :: "'f" shows "run_state (x \<bind> f) s = (case run_state x s of (a, s') \<Rightarrow> run_state (f a) s')"
integral_sin: fixes a :: "real"   and b :: "real" assumes "a \<le> b" shows "integral {a..b} sin = cos a - cos b"
w_simps_3: shows "src f.chine = src (tab\<^sub>0 f)"
rai': shows "real_alg_2' ri q l r = real_alg_2_main ri (q, l', r')"
one_dim_norm: fixes x :: "'b" shows "norm x = cmod (one_dim_iso x)"
in_hom_char: fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "('a, 'b) concrete_category.arr"   and a :: "('a, 'b) concrete_category.arr"   and b :: "('a, 'b) concrete_category.arr" assumes "concrete_category Obj Hom Id Comp" shows "partial_magma.in_hom (concrete_category.COMP Obj Hom Comp) f a b = (partial_magma.arr (concrete_category.COMP Obj Hom Comp) f \<and> partial_magma.ide (concrete_category.COMP Obj Hom Comp) a \<and> partial_magma.ide (concrete_category.COMP Obj Hom Comp) b \<and> concrete_category.Dom f = concrete_category.Dom a \<and> Cod f = concrete_category.Dom b)"
antisym: fixes x :: "'b"   and y :: "'b" assumes "ord' x y"   and "ord' y x" shows "x = y"
Comm1: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi"   and Q :: "pi"   and Q' :: "pi" assumes "P \<Longrightarrow>a<b> \<prec> P'"   and "Q \<Longrightarrow>a[b] \<prec> Q'" shows "P \<parallel> Q \<Longrightarrow>\<tau> \<prec> P' \<parallel> Q'"
outgoing_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and v :: "nat" shows "Graph.outgoing c v \<equiv> {(v, u) |u. (v, u) \<in> Graph.E c}"
ex_norm1: assumes "UNIV \<noteq> {0::'a}" shows "\<exists>x. norm x = 1"
redT_elims: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "s -t\<triangleright>ta\<rightarrow> s'" obtains x :: "'x" and x' :: "'x" and m' :: "'m" and ws' :: "'t \<Rightarrow> 'w wait_set_status option" where "t \<turnstile> \<langle>x__, shr s\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>" and "thr s t = \<lfloor>(x__, no_wait_locks)\<rfloor>" and "lock_ok_las (locks s) t \<lbrace>ta\<rbrace>\<^bsub>l\<^esub>" and "thread_oks (thr s) \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>" and "cond_action_oks s t \<lbrace>ta\<rbrace>\<^bsub>c\<^esub>" and "wset_actions_ok (wset s) t \<lbrace>ta\<rbrace>\<^bsub>w\<^esub>" and "interrupt_actions_ok (interrupts s) \<lbrace>ta\<rbrace>\<^bsub>i\<^esub>" and "redT_updWs t (wset s) \<lbrace>ta\<rbrace>\<^bsub>w\<^esub> ws'" and "s' = (redT_updLs (locks s) t \<lbrace>ta\<rbrace>\<^bsub>l\<^esub>, (redT_updTs (thr s) \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>(t \<mapsto> (x', redT_updLns (locks s) t no_wait_locks \<lbrace>ta\<rbrace>\<^bsub>l\<^esub>)), m'), ws', redT_updIs (interrupts s) \<lbrace>ta\<rbrace>\<^bsub>i\<^esub>)"   | x :: "'x" and ln :: "'l \<Rightarrow>f nat" and n :: "'l" where "thr s t = \<lfloor>(x, ln)\<rfloor>" and "ta = (K$ [], [], [], [], [], convert_RA ln)" and "\<not> waiting (wset s t)" and "may_acquire_all (locks s) t ln" and "0 < ln $ n" and "s' = (acquire_all (locks s) t ln, (thr s(t \<mapsto> (x, no_wait_locks)), shr s), wset s, interrupts s)"
CHAOS_def: fixes A :: "'b set" shows "CHAOS A \<equiv> \<mu> x. STOP |-| Mprefix A (\<lambda>xa. x)"
mul_instr_sub1_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and instr_name :: "sparc_operation"   and result :: "word32"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = snd (fst (mul_instr_sub1 instr_name result s1)) \<and> t2 = snd (fst (mul_instr_sub1 instr_name result s2))" shows "low_equal t1 t2"
aff_dim_halfspace_le: fixes a :: "'a"   and r :: "real" shows "aff_dim {x. a \<bullet> x \<le> r} = (if a = (0::'a) \<and> r < 0 then - 1 else int DIM('a))"
le__nlt: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Le C D" shows "\<not> C D Lt A B"
seq_annil: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and seq :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and seqstar :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.cka plus less_eq less one zero times star seq seqstar" shows "seq zero x = zero"
rotate12: fixes A12 :: "fm"   and A1 :: "fm"   and A2 :: "fm"   and A3 :: "fm"   and A4 :: "fm"   and A5 :: "fm"   and A6 :: "fm"   and A7 :: "fm"   and A8 :: "fm"   and A9 :: "fm"   and A10 :: "fm"   and A11 :: "fm"   and H :: "fm set"   and B :: "fm" assumes "insert A12 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 H))))))))))) \<turnstile> B" shows "insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 (insert A12 H))))))))))) \<turnstile> B"
relS_def: fixes M :: "('b \<Rightarrow> 'b ref) set" shows "relS M = \<Union> (rel ` M)"
ODord_less_not_eq: fixes x :: "'c Order set"   and y :: "'c Order set" assumes "x \<in> carrier ODnods"   and "y \<in> carrier ODnods"   and "x \<sqsubset> y" shows "x \<noteq> y"
LeftDerivationIntro_cut_prefix: fixes \<delta> :: "symbol list"   and \<alpha> :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and ix :: "nat"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and j :: "nat"   and \<gamma> :: "symbol list" assumes "LeftDerivationIntro (\<delta> @ \<alpha>) i r ix D j \<gamma>"   and "derivation_ge D (length \<delta>)"   and "length \<delta> \<le> i"   and "is_word \<delta>" shows "\<exists>\<gamma>'. \<gamma> = \<delta> @ \<gamma>' \<and> LeftDerivationIntro \<alpha> (i - length \<delta>) r ix (derivation_shift D (length \<delta>) 0) (j - length \<delta>) \<gamma>'"
first_tail_second: fixes s :: "nat \<Rightarrow> 'b" shows "first (tail s) = second s"
tranclp_rtranclp_absorb: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "R\<^sup>+\<^sup>+\<^sup>*\<^sup>* = R\<^sup>*\<^sup>*"