hcju/isabelleps huggingface dataset:
Bseq_eventually_mono: fixes f :: "nat \<Rightarrow> 'e"   and g :: "nat \<Rightarrow> 'f" assumes "\<forall>\<^sub>F n in sequentially. norm (f n) \<le> norm (g n)"   and "Bseq g" shows "Bseq f"
progress_Until_le: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and \<phi> :: "'a MFOTL.formula"   and I :: "\<I>"   and \<psi> :: "'a MFOTL.formula"   and j :: "nat" shows "Monitor.progress \<sigma> (formula.Until \<phi> I \<psi>) j \<le> min (Monitor.progress \<sigma> \<phi> j) (Monitor.progress \<sigma> \<psi> j)"
cases: fixes a :: "('c, unit) Old_Datatype.node set" assumes "a \<in> sexp" obtains aa :: "'c" where "a = Leaf aa"   | i :: "nat" where "a = Numb i"   | M :: "('c, unit) Old_Datatype.node set" and N :: "('c, unit) Old_Datatype.node set" where "a = Scons M N" and "M \<in> sexp" and "N \<in> sexp"
submap_imp_state_succ_submap_a: fixes s1 :: "('d, 'e) fmap"   and s2 :: "('d, 'e) fmap"   and s3 :: "('d, 'e) fmap" assumes "s1 \<subseteq>\<^sub>f s2"   and "s2 \<subseteq>\<^sub>f s3" shows "s1 \<subseteq>\<^sub>f s3"
sum_not_less_1: 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 b :: "'a"   and a :: "'a" assumes "class.weight plus zero less_eq less"   and "less_eq zero b" shows "\<not> less (plus a b) a"
success_sensitive_iff_source_target_rel_respects_success: fixes success :: "'barbs" shows "((\<forall>S x. x \<in> {success} \<and> ((\<exists>Sa. Sa \<in>S SourceTerm S \<and> Sa\<down><SWB>x) \<or> (\<exists>T. T \<in>T SourceTerm S \<and> T\<down><TWB>x)) \<longrightarrow> x \<in> {success} \<and> ((\<exists>Sa. Sa \<in>S TargetTerm (\<lbrakk>S\<rbrakk>) \<and> Sa\<down><SWB>x) \<or> (\<exists>T. T \<in>T TargetTerm (\<lbrakk>S\<rbrakk>) \<and> T\<down><TWB>x))) \<and> (\<forall>S x. x \<in> {success} \<and> ((\<exists>Sa. Sa \<in>S TargetTerm (\<lbrakk>S\<rbrakk>) \<and> Sa\<down><SWB>x) \<or> (\<exists>T. T \<in>T TargetTerm (\<lbrakk>S\<rbrakk>) \<and> T\<down><TWB>x)) \<longrightarrow> x \<in> {success} \<and> ((\<exists>Sa. Sa \<in>S SourceTerm S \<and> Sa\<down><SWB>x) \<or> (\<exists>T. T \<in>T SourceTerm S \<and> T\<down><TWB>x)))) = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> rel_respects_barb_set Rel (STCalWB SWB TWB) {success})"
witness_3: shows "t' \<in>\<^sub>A frontier (c_pts c loc') \<or> zcount (c_work c loc') t' < 0"
vpa_Main_slpa: fixes cs :: "'edge list"   and as :: "'edge list"   and m :: "'node"   and m' :: "'node" assumes "valid_path_aux cs as"   and "m -as\<rightarrow>* m'"   and "as \<noteq> []"   and "valid_call_list cs m"   and "get_proc m' = Main"   and "get_proc (case cs of [] \<Rightarrow> m | a # list \<Rightarrow> sourcenode (last cs)) = Main" shows "same_level_path_aux cs as \<and> upd_cs cs as = []"
subgroups_Inter: fixes A :: "'a set set" assumes "\<And>H. H \<in> A \<Longrightarrow> subgroup H G"   and "A \<noteq> {}" shows "subgroup (\<Inter> A) G"
inv_solve_right: fixes a :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and b :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and c :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set" assumes "a \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))"   and "b \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))"   and "c \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))" shows "(a = b \<otimes>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> inv\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> c) = (b = a \<otimes>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> c)"
kle: fixes u :: "complex" assumes "cmod (u - z) = r" shows "k \<le> cmod (u - w)"
elims_3: fixes x :: "'a \<times> 'b \<times> 'c"   and xa :: "'a \<times> 'd"   and xb :: "'d \<Rightarrow> 'b"   and xc :: "'a \<Rightarrow> 'd \<Rightarrow> 'c" assumes "\<not> triple_eq_pair x xa xb xc" obtains t :: "'a" and a1 :: "'b" and a2 :: "'c" and ts' :: "'a" and tp' :: "'d" and f :: "'d \<Rightarrow> 'b" and g :: "'a \<Rightarrow> 'd \<Rightarrow> 'c" where "x = (t, a1, a2)" and "xa = (ts', tp')" and "xb = f" and "xc = g" and "\<not> (t = ts' \<and> a1 = f tp' \<and> a2 = g ts' tp')"
polyfun_const: fixes N :: "nat set"   and c :: "'b" shows "polyfun N (\<lambda>_. c)"
nextFreeBlockInvariant1: shows "nextFreeBlockInvariant makeCF"
assms_2: shows "\<Psi> \<rhd> Q \<doteq> R"
norm_less_1_bounded: shows "bdd_above {\<parallel>f *\<^sub>v x\<parallel> |x. \<parallel>x\<parallel> < 1}"
varset_act_diff_un_imp_varset_diff: fixes a :: "('c, bool) fmap \<times> ('c, bool) fmap"   and vs'' :: "'c set"   and vs' :: "'c set"   and vs :: "'c set" assumes "varset_action a (vs'' - (vs' \<union> vs))" shows "varset_action a (vs'' - vs)"
span_closed_sum_list: fixes xs :: "'a list"   and B :: "'a set" assumes "\<And>x. x \<in> set xs \<Longrightarrow> x \<in> ideal B" shows "sum_list xs \<in> ideal B"
quadratic_sub_less: fixes var :: "nat"   and L :: "real list"   and Dv :: "real"   and Cv :: "real"   and c :: "real mpoly"   and a :: "real mpoly"   and Av :: "real"   and b :: "real mpoly"   and Bv :: "real"   and d :: "real mpoly"   and p :: "real mpoly" assumes "var < length L"   and "Dv \<noteq> 0"   and "0 \<le> Cv"   and "var \<notin> vars c"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) a = Av"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) b = Bv"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) c = Cv"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) d = Dv" shows "aEval (Less p) (L[var := (Av + Bv * sqrt Cv) / Dv]) = eval (quadratic_sub var a b c d (Less p)) (L[var := sqrt Cv])"
that: fixes k' :: "nat" assumes "n' cd\<^bsup>\<pi>'\<^esup>\<rightarrow> k'"   and "cs\<^bsup>\<pi>\<^esup> k = cs\<^bsup>\<pi>'\<^esup> k'" shows "thesis"
D: shows "dfa_steps (min_dfa M) 0 w = fst (mk_eqcl (replicate (length (fst M)) None) [] 0 (fixpt M (init_tr M))) ! dfa_steps M 0 w"
lnth_0: fixes x :: "'a"   and xs :: "'a llist" shows "lnth (LCons x xs) 0 = x"
balL_def: fixes AB :: "('b \<times> nat) tree"   and c :: "'b"   and C :: "('b \<times> nat) tree" shows "balL AB c C = (if ht AB = ht C + 2 then case AB of \<langle>A, (a, x), B\<rangle> \<Rightarrow> if ht B \<le> ht A then node A a (node B c C) else case B of \<langle>B\<^sub>1, (b, x), B\<^sub>2\<rangle> \<Rightarrow> node (node A a B\<^sub>1) b (node B\<^sub>2 c C) else node AB c C)"
conv_abscissa_minus: fixes f :: "'a fds" shows "conv_abscissa (- f) = conv_abscissa f"
outside_mono: fixes S :: "'a set"   and T :: "'a set" assumes "S \<subseteq> T" shows "outside T \<subseteq> outside S"
Int_Un_distrib2: fixes B :: "'d set"   and C :: "'d set"   and A :: "'d set" shows "(B \<union> C) \<inter> A = B \<inter> A \<union> C \<inter> A"
intros_1: fixes S :: "'a spmf \<Rightarrow> 'b spmf \<Rightarrow> bool"   and p :: "'a spmf"   and q :: "'b spmf" assumes "S p q" shows "trace_eq_simcl S p q"
lemma_9_3: fixes d :: "'a \<Rightarrow> nat"   and rword :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and G :: "('t \<Rightarrow>\<^sub>0 'b) set"   and u :: "'t"   and g1 :: "'t \<Rightarrow>\<^sub>0 'b"   and h :: "'t \<Rightarrow>\<^sub>0 'b" assumes "dickson_grading d"   and "is_rewrite_ord rword"   and "is_RB_upt d rword G u"   and "inj_on lt G"   and "\<not> is_syz_sig d u"   and "is_canon_rewriter rword G u g1"   and "h \<in> G"   and "is_sig_red (\<prec>\<^sub>t) (=) {h} (monom_mult (1::'b) (pp_of_term u - lp g1) g1)"   and "d (pp_of_term u) \<le> dgrad_max d" shows "is_regular_spair g1 h"
update_update_1: fixes k :: "'b"   and v :: "'c"   and w :: "'c"   and m :: "('b, 'c) mapping" shows "update k v (update k w m) = update k v m"
func: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set set"   and T :: "'c set set" assumes "func f S T" shows "f \<in> carrier\<^bsub>S\<^esub> \<rightarrow> carrier\<^bsub>T\<^esub>"
simps_7: fixes p :: "'e" shows "nothrows (Call p) = True"
kmax_prop: fixes f :: "nat \<Rightarrow> real" shows "kmax f PR \<subseteq> PR \<and> card (kmax f PR) = khl \<and> (\<forall>i\<in>kmax f PR. \<forall>j\<in>PR - kmax f PR. f j \<le> f i)"
Comp: fixes Env :: "env"   and B :: "lname set"   and c1 :: "stmt"   and C1 :: "assigned"   and c2 :: "stmt"   and C2 :: "assigned"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>c1\<rangle>\<guillemotright> C1"   and "Env\<turnstile> nrm C1 \<guillemotright>\<langle>c2\<rangle>\<guillemotright> C2"   and "nrm A = nrm C2"   and "brk A = brk C1 \<Rightarrow>\<inter>  brk C2" shows "Env\<turnstile> B \<guillemotright>\<langle>c1;; c2\<rangle>\<guillemotright> A"
convex_hull_caratheodory_aff_dim: fixes p :: "'a set" shows "convex hull p = {y. \<exists>S u. finite S \<and> S \<subseteq> p \<and> int (card S) \<le> aff_dim p + 1 \<and> (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = y}"
quot_domain_iff_primeideal: fixes P :: "'a set" assumes "ideal P R" shows "domain (R Quot P) = primeideal P R"
size_q: shows "size (queue_to_multiset (xs @ [x])) = size (queue_to_multiset xs) + size (tree_to_multiset x)"
cod_char: fixes f :: "('a, 'a set) Maps.arr" shows "Maps.cod f = (if Maps.arr f then Maps.MkIde (Maps.Cod f) else Maps.null)"
prems_1: fixes i :: "'a ifex"   and t :: "'a ifex"   and e :: "'a ifex" assumes "lowest_tops [i, t, e] = None"   and "i = Trueif" shows "P i t e"
cdf_is_right_cont: fixes a :: "real" shows "continuous (at_right a) (cdf M2)"
simps: fixes q :: "'d"   and l :: "'e"   and qs :: "'d list" shows "rhsq q \<rightarrow> l qs = qs"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) (A_B.Dom (fst Fg)) (A_B.Cod (fst Fg')) F'oF.map"
vrange_atD: fixes r :: "V"   and b :: "V" assumes "vsv r"   and "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r" shows "\<exists>a\<in>elts (\<D>\<^sub>\<circ> r). r\<lparr>a\<rparr> = b"
sep_add_disjI2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x + y ## z"   and "x ## y" shows "z + y ## x"
induct: fixes P :: "('d, 'a, 'b) sp\<^sub>\<mu> \<Rightarrow> ('c, 'd, ('c, 'd) sp\<^sub>\<nu>) sp\<^sub>\<mu> \<Rightarrow> bool"   and a0 :: "('d, 'a, 'b) sp\<^sub>\<mu>"   and a1 :: "('c, 'd, ('c, 'd) sp\<^sub>\<nu>) sp\<^sub>\<mu>" assumes "\<And>b sp fsp. P (Put b sp) fsp"   and "\<And>f b sp. P (f b) (out sp) \<Longrightarrow> P (Get f) (Put b sp)"   and "\<And>f g. (\<And>x. P (Get f) (g x)) \<Longrightarrow> P (Get f) (Get g)" shows "P a0 a1"
invert_two_sides: fixes A :: "'p"   and B :: "'p"   and P :: "'p"   and Q :: "'p" assumes "A B TS P Q" shows "B A TS P Q"
partial_order_on_relation_ofI: fixes A :: "'c set"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>a. a \<in> A \<Longrightarrow> P a a"   and "\<And>a b c. \<lbrakk>a \<in> A; b \<in> A; c \<in> A; P a b; P b c\<rbrakk> \<Longrightarrow> P a c"   and "\<And>a b. \<lbrakk>a \<in> A; b \<in> A; P a b; P b a\<rbrakk> \<Longrightarrow> a = b" shows "partial_order_on A (relation_of P A)"
mtx_invertibleI: fixes A :: "'a sq_mtx"   and B :: "'a sq_mtx" assumes "A * B = 1"   and "B * A = 1" shows "mtx_invertible A"
inext_nth_atLeastAtMost: fixes a :: "nat"   and n :: "nat"   and m :: "nat" assumes "a \<le> n - m"   and "m \<le> n" shows "{m..n} \<rightarrow> a = m + a"
less_three: fixes n :: "nat" shows "(n < Suc (Suc (Suc 0))) = (n = 0 \<or> n = 1 \<or> n = 2)"
in_bd: fixes A1 :: "'d set"   and A2 :: "'e set"   and A3 :: "'f set" shows "|F1in A1 A2 A3| \<le>o ((|A1| +c |A2| +c |A3|) +c ctwo) ^c F1bd'"
vec_eq_iff: fixes x :: "'b Matrix.vec"   and y :: "'b Matrix.vec" shows "(x = y) = (dim_vec x = dim_vec y \<and> (\<forall>i<dim_vec y. x $v i = y $v i))"
intro: fixes cnf :: "lit uprod set" assumes "finite (vars_of_cnf cnf)"   and "\<nexists>x. (Pos x, Neg x) \<in> (imp_graph cnf)\<^sup>* \<and> (Neg x, Pos x) \<in> (imp_graph cnf)\<^sup>*"   and "is_2sat cnf" shows "construct_sa cnf"
Liminf_le_Limsup: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b" assumes "F \<noteq> \<bottom>" shows "Liminf F f \<le> Limsup F f"
ker_in_span: shows "{x. f x = (0::'c)} \<subseteq> local.span B"
IntE: fixes c :: "'g"   and A :: "'g set"   and B :: "'g set" assumes "c \<in> A \<inter> B" obtains "c \<in> A" and "c \<in> B"
disc_6: fixes x31 :: "'c"   and x32 :: "'d" shows "is_Yield (Yield x31 x32)"
add_pow_int_ge: fixes k :: "int"   and R :: "('c, 'd) ring_scheme"   and a :: "'c" assumes "0 \<le> k" shows "[k] \<cdot>\<^bsub>R\<^esub> a = [nat k] \<cdot>\<^bsub>R\<^esub> a"
comp_arr_ide: fixes a :: "'b"   and f :: "'b" assumes "\<phi>.B.ide a"   and "\<phi>.B.seq f a" shows "f \<cdot>\<^sub>B a = f"
in_set_conv_decomp: fixes x :: "'b"   and xs :: "'b list" shows "(x \<in> set xs) = (\<exists>ys zs. xs = ys \<cdot> x # zs)"
connected_component_trans: fixes S :: "'b set"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "connected_component S x y"   and "connected_component S y z" shows "connected_component S x z"
INT_insert: fixes B :: "'d \<Rightarrow> 'c set"   and a :: "'d"   and A :: "'d set" shows "\<Inter> (B ` insert a A) = B a \<inter> \<Inter> (B ` A)"
Theorem_1: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and ds :: "'e set" assumes "ContractsWithBilateralSubstitutesAndIRC Xd Xh Pd Ch" shows "Contracts.stable_on Pd Ch ds (Contracts.CH Ch (Contracts.fp_cop_F Pd Ch ds))"
det_multrow_mat: fixes k :: "nat"   and n :: "nat"   and a :: "'b" assumes "k < n" shows "det (multrow_mat n k a) = a"
simps: fixes a1 :: "'node SDG_node"   and a2 :: "'node SDG_node list"   and a3 :: "'node SDG_node" shows "a1 i-a2\<rightarrow>\<^sub>d* a3 = ((\<exists>n. a1 = n \<and> a2 = [] \<and> a3 = n \<and> valid_SDG_node n) \<or> (\<exists>n ns n'' n'. a1 = n \<and> a2 = ns @ [n''] \<and> a3 = n' \<and> n i-ns\<rightarrow>\<^sub>d* n'' \<and> n'' \<longrightarrow>\<^bsub>cd\<^esub> n') \<or> (\<exists>n ns n'' V n'. a1 = n \<and> a2 = ns @ [n''] \<and> a3 = n' \<and> n i-ns\<rightarrow>\<^sub>d* n'' \<and> n'' -V\<rightarrow>\<^sub>d\<^sub>d n' \<and> n'' \<noteq> n'))"
permute_finite: fixes p :: "perm"   and X :: "'e set" shows "finite (p \<bullet> X) = finite X"
gseq_def: shows "gseq = {(f, g). f \<in> SEQ trees \<and> g \<in> SEQ trees \<and> (\<exists>i. size (g i) < size (f i) \<and> (\<forall>j<i. f j = g j))}"
spr_repRels_def: fixes envObs :: "'b \<Rightarrow> ('b, 'c, 'd) BEState \<Rightarrow> 'e \<times> 'd option" shows "spr_repRels envObs \<equiv> \<lambda>a. rel_ext (\<lambda>((u, v), u', v'). envObs a u = envObs a u' \<and> envObs a v = envObs a v')"
butlast_strict_prefix_eq_butlast: fixes s :: "'b list"   and t :: "'b list" assumes "length s = length t"   and "strict_prefix (butlast s) t" shows "strict_prefix (butlast s) t = (butlast s = butlast t)"
leof_use_spec_rule: fixes m :: "'a nres"   and \<Psi> :: "'a \<Rightarrow> bool"   and \<Phi> :: "'a \<Rightarrow> bool" assumes "m \<le>\<^sub>n SPEC \<Psi>"   and "m \<le>\<^sub>n SPEC (\<lambda>s. \<Psi> s \<longrightarrow> \<Phi> s)" shows "m \<le>\<^sub>n SPEC \<Phi>"
norm_7: fixes p :: "'g \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'g"   and q :: "'g \<Rightarrow> bool" shows "(\<lceil>p\<rceil> \<sigma> \<and> \<lceil>q\<rceil> \<sigma>) = \<lceil>\<lambda>s. p s \<and> q s\<rceil> \<sigma>"
rep_number_on_set_def: fixes B :: "'b set set"   and x :: "'b" assumes "finite B" shows "mset_set B rep x = int (card {b \<in> B. x \<in> b})"
eval_cases_1: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and C :: "char list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "P,E \<turnstile> \<langle>new C,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" obtains h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" and a :: "nat" and l :: "char list \<Rightarrow> val option" where "s = (h__, l__)" and "e' = ref (a, [C])" and "s' = (h__(a \<mapsto> (C, Collect (init_obj P C))), l__)" and "new_Addr h__ = \<lfloor>a\<rfloor>"   | h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" and l :: "char list \<Rightarrow> val option" where "s = (h, l)" and "e' = Throw (addr_of_sys_xcpt OutOfMemory, [OutOfMemory])" and "s' = (h, l)" and "new_Addr h = None"
square_def: fixes A :: "state \<times> state \<Rightarrow> bool"   and v :: "state \<Rightarrow> 'a" shows "[A]_v \<equiv> A \<or> unchanged v"
LDF': shows "LeftDerivation (b1 @ [X] @ b2) D c \<and> length b1 < length (b1 @ [X] @ b2) \<and> j < length c \<and> (b1 @ [X] @ b2) ! length b1 = c ! j \<and> (\<exists>E F. D = E @ derivation_shift F 0 (Suc j) \<and> LeftDerivation (take (length b1) (b1 @ [X] @ b2)) E (take j c) \<and> LeftDerivation (drop (Suc (length b1)) (b1 @ [X] @ b2)) F (drop (Suc j) c))"
map_comp0: fixes g1 :: "'j \<Rightarrow> 'g"   and f1 :: "'d \<Rightarrow> 'j"   and g2 :: "'k \<Rightarrow> 'h"   and f2 :: "'e \<Rightarrow> 'k"   and g3 :: "'l \<Rightarrow> 'i"   and f3 :: "'f \<Rightarrow> 'l" shows "F2map (g1 \<circ> f1) (g2 \<circ> f2) (g3 \<circ> f3) = F2map g1 g2 g3 \<circ> F2map f1 f2 f3"
not_cond: fixes P :: "'b \<times> 'b \<Rightarrow> bool"   and b :: "'b \<times> 'b \<Rightarrow> bool"   and Q :: "'b \<times> 'b \<Rightarrow> bool" shows "(\<not> (P \<triangleleft> b \<triangleright>  Q)) = (\<not> P \<triangleleft> b \<triangleright>  \<not> Q)"
invariant_includes_reachable: fixes F :: "'c program"   and A :: "'c set" assumes "F \<in> invariant A" shows "reachable F \<subseteq> A"
update_convs_6: fixes cache' :: "cpu_cache \<Rightarrow> cpu_cache"   and cpu_reg :: "CPU_register \<Rightarrow> word32"   and user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and sys_reg :: "sys_reg \<Rightarrow> word32"   and mem :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option"   and mmu :: "MMU_state"   and cache :: "cpu_cache"   and dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and state_var :: "sparc_state_var"   and traps :: "Trap set"   and undef :: "bool"   and more :: "'c" shows "cache_update cache' \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr> = \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache' cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr>"
assms_1: fixes A :: "'a set" shows "Sigma_Algebra.measure (measure_spmf p) A \<le> Sigma_Algebra.measure (measure_spmf q) {y. \<exists>x\<in>A. R x y}"
simps_1: fixes t :: "'b tree234" shows "tree\<^sub>i (T\<^sub>i t) = t"
zero: shows "R (arith_ops_record.zero ops) (0::'a)"
Iset_star_n: fixes X :: "nat \<Rightarrow> 'c"   and A :: "nat \<Rightarrow> 'c set" shows "(star_n X \<in> Iset (star_n A)) = (\<forall>\<^sub>F n in \<U>. X n \<in> A n)"
prems_2: shows "\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> =n\<Rightarrow> Fault f"
of_match_fields_safe_eq2: fixes m :: "of_match_field set"   and p :: "(32, 'b) simple_packet_ext_scheme" assumes "all_prerequisites m" shows "OF_match_fields_safe m p = (OF_match_fields m p = Some True)"
the_gpv_parametric': fixes A :: "'g \<Rightarrow> 'j \<Rightarrow> bool"   and C :: "'h \<Rightarrow> 'k \<Rightarrow> bool"   and R :: "'i \<Rightarrow> 'l \<Rightarrow> bool" shows "(rel_gpv'' A C R ===> rel_spmf (rel_generat A C (R ===> rel_gpv'' A C R))) the_gpv the_gpv"
bind_map_spmf: fixes f :: "'e \<Rightarrow> 'd"   and p :: "'e spmf"   and g :: "'d \<Rightarrow> 'b spmf" shows "map_spmf f p \<bind> g = p \<bind> g \<circ> f"
cases: fixes x :: "'b memref list" obtains "x = []"   | r :: "'b memref" and rs :: "'b memref list" where "x = r # rs"
intros_7: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c\<^sub>2 :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate" shows "\<Gamma>\<turnstile> (Seq Skip c\<^sub>2, s) \<rightarrow> (c\<^sub>2, s)"
collapse_3: fixes stateful_strand_step :: "('d, 'e) stateful_strand_step" assumes "stateful_strand_step.is_Equality stateful_strand_step" shows "\<langle>the_check stateful_strand_step: the_lhs stateful_strand_step \<doteq> the_rhs stateful_strand_step\<rangle> = stateful_strand_step"
VU: fixes y :: "'c" assumes "y \<in> U" shows "V y \<subseteq> U"
direct_path_return: fixes x :: "'n" assumes "x \<in> local.nodes"   and "x \<noteq> local.return" obtains \<pi> :: "nat \<Rightarrow> 'n" and n :: "nat" where "is_path \<pi>" and "\<pi> 0 = x" and "\<pi> n = local.return" and "\<forall>i>0. \<pi> i \<noteq> x"
t_fourtimes_change_term_state: fixes ires :: "cell list"   and rs :: "nat"   and n :: "nat" shows "\<exists>stp ln rn. steps0 (Suc 0, Bk # Bk # ires, Oc \<up> Suc rs @ Bk \<up> n) t_fourtimes stp = (Suc t_fourtimes_len, Bk \<up> ln @ Bk # Bk # ires, Oc \<up> Suc (4 * rs) @ Bk \<up> rn)"
Proposition_1: fixes n :: "nat" assumes "\<And>f xs. \<lbrakk>associative f; xs \<noteq> []\<rbrakk> \<Longrightarrow> candidate f xs = scanl1 f xs" shows "candidate (@) (map wrap [0..<n + 1]) = ups n"
compl_unique: fixes x :: "'b upred"   and y :: "'b upred" assumes "(x \<and> y) = false"   and "(x \<or> y) = true" shows "(\<not> x) = y"
equivE: fixes A :: "'e set"   and r :: "('e \<times> 'e) set" assumes "equiv A r" obtains "refl_on A r" and "sym r" and "trans r"
tendsto_add_ereal_general: fixes x :: "ereal"   and y :: "ereal"   and f :: "'a \<Rightarrow> ereal"   and F :: "'a filter"   and g :: "'a \<Rightarrow> ereal" assumes "\<not> (x = \<infinity> \<and> y = - \<infinity> \<or> x = - \<infinity> \<and> y = \<infinity>)"   and "(f \<longlongrightarrow> x) F"   and "(g \<longlongrightarrow> y) F" shows "((\<lambda>x. f x + g x) \<longlongrightarrow> x + y) F"
amin_le1: fixes z :: "ant"   and z' :: "ant"   and w :: "ant" assumes "z \<le> z'" shows "amin z w \<le> z'"
content_cbox_plus: fixes x :: "'a"   and h :: "real" shows "content (cbox x (x + h *\<^sub>R One)) = (if 0 \<le> h then h ^ DIM('a) else 0)"
weak_consensus_def: fixes vals :: "'a \<Rightarrow> 'b"   and dec :: "'c \<Rightarrow> 'b option"   and rho :: "'d \<Rightarrow> 'e \<Rightarrow> 'c" shows "weak_consensus vals dec rho \<equiv> (\<forall>v. (\<forall>p. vals p = v) \<longrightarrow> (\<forall>n p w. dec (rho n p) = Some w \<longrightarrow> w = v)) \<and> (\<forall>m n p q v w. dec (rho m p) = Some v \<and> dec (rho n q) = Some w \<longrightarrow> v = w) \<and> (\<forall>p. \<exists>n. dec (rho n p) \<noteq> None)"
pcp_sat: fixes C :: "'b formula set set"   and S :: "'b formula set" assumes "pcp (C::'b formula set set)"   and "(S::'b formula set) \<in> (C::'b formula set set)" shows "sat (S::'b formula set)"
inject_5: fixes x5 :: "nat"   and y5 :: "nat" shows "(Key x5 = Key y5) = (x5 = y5)"
rel_cong: fixes x :: "'e option"   and ya :: "'e option"   and y :: "'f option"   and xa :: "'f option"   and R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and Ra :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "x = ya"   and "y = xa"   and "\<And>z yb. \<lbrakk>z \<in> set_option ya; yb \<in> set_option xa\<rbrakk> \<Longrightarrow> R z yb = Ra z yb" shows "rel_option R x y = rel_option Ra ya xa"