hcju/isabelleps huggingface dataset:
\<tau>Exec_methr_step': fixes addr2thread_id :: "'a \<Rightarrow> 'b"   and thread_id2addr :: "'b \<Rightarrow> 'a"   and spurious_wakeups :: "bool"   and empty_heap :: "'c"   and allocate :: "'c \<Rightarrow> htype \<Rightarrow> ('c \<times> 'a) set"   and typeof_addr :: "'c \<Rightarrow> 'a \<Rightarrow> htype option"   and heap_read :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> bool"   and heap_write :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> 'c \<Rightarrow> bool"   and ci :: "('a, 'c) check_instr"   and P :: "(nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and ins :: "'a instr list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "'b"   and h :: "'c"   and s :: "'a val list \<times> 'a val list \<times> nat \<times> 'a option"   and stk' :: "'a val list"   and loc' :: "'a val list"   and pc' :: "nat"   and xcp' :: "'a option"   and s' :: "'a val list \<times> 'a val list \<times> nat \<times> 'a option" assumes "(JVM_heap_base.\<tau>exec_meth addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P ins xt t h)\<^sup>*\<^sup>* s (stk', loc', pc', xcp')"   and "JVM_heap_base.\<tau>exec_meth addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P ins xt t h (stk', loc', pc', xcp') s'" shows "(JVM_heap_base.\<tau>exec_meth addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P ins xt t h)\<^sup>*\<^sup>* s s'"
assms: shows "a \<in>\<^sub>\<circ> \<omega>"
assms_2: shows "validate_clause_set I (cl_ecl ` P)"
scaleRe_ivl_coll_impl_def: fixes li :: "ereal"   and ui :: "ereal"   and Xi :: "'d list" shows "local.scaleRe_ivl_coll_impl li ui Xi \<equiv> dFORWEAK_LIST Xi (dRETURN []) (\<lambda>xa. (if li < ui \<or> - \<infinity> < li \<and> li \<le> ui \<and> ui < \<infinity> then dRETURN ((li, ui), xa) else dSUCCEED) \<bind> (\<lambda>xb. dRETURN [xb])) (\<lambda>xa y. dRETURN (y @ xa))"
simps_4: fixes b :: "'b set"   and x :: "'b set"   and y :: "'b \<Rightarrow> nat"   and c :: "'b com"   and s :: "'b"   and s' :: "'b" shows "Sem (While b x y c) s s' = (\<exists>n. iter n b (Sem c) s s')"
assms_1: shows "h \<turnstile> f \<rightarrow>\<^sub>e e"
extends_memb: fixes \<Delta> :: "binding list"   and \<Gamma> :: "binding list"   and X :: "tyvrs"   and T :: "ty" assumes "\<Delta> extends \<Gamma>"   and "TVarB X T \<in> set \<Gamma>" shows "TVarB X T \<in> set \<Delta>"
intros_3: fixes f :: "'d"   and \<phi> :: "'b" shows "hereditarily_fs (Pred\<^sub>\<alpha> f \<phi>)"
P''TransP''': shows "P'' \<longmapsto> \<tau> \<prec> P'''"
in_hhom_def: fixes \<mu> :: "'a"   and a :: "'a"   and b :: "'a" shows "\<guillemotleft>\<mu> : a \<rightarrow> b\<guillemotright> \<equiv> arr \<mu> \<and> src \<mu> = a \<and> trg \<mu> = b"
resPres: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and x :: "name" assumes "P \<leadsto>[Rel] Q"   and "\<And>R S y. (R, S) \<in> Rel \<Longrightarrow> (<\<nu>y>R, <\<nu>y>S) \<in> Rel'"   and "Rel \<subseteq> Rel'"   and "eqvt Rel"   and "eqvt Rel'" shows "<\<nu>x>P \<leadsto>[Rel'] <\<nu>x>Q"
npGTPowZero: fixes x :: "'a"   and m :: "nat"   and n :: "nat" assumes "x \<in> carrier R"   and "x^\<^bsup>R m\<^esup> = \<zero>"   and "m \<le> n" shows "x^\<^bsup>R n\<^esup> = \<zero>"
subst_atm_mset_list_tl: fixes AAs :: "'a multiset list"   and \<sigma> :: "'s" shows "tl (AAs \<cdot>aml \<sigma>) = tl AAs \<cdot>aml \<sigma>"
do_uncurry: fixes f :: "'f \<Rightarrow> 'g \<Rightarrow> 'e"   and a :: "'f"   and b :: "'g" shows "f a b = uncurry f (a, b)"
freshVar_def: fixes vs :: "vbl set" shows "freshVar vs = X (LEAST n. n \<notin> deX ` vs)"
either_height_gt_0_imp_sibling: fixes t\<^sub>1 :: "'a tree"   and t\<^sub>2 :: "'a tree"   and w :: "nat"   and a :: "'a" assumes "0 < height t\<^sub>1 \<or> 0 < height t\<^sub>2" shows "sibling (Node w t\<^sub>1 t\<^sub>2) a = (if a \<in> alphabet t\<^sub>1 then sibling t\<^sub>1 a else sibling t\<^sub>2 a)"
restrict_current_def: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" shows "g \<upharpoonleft> \<Gamma> / f = (\<lambda>(x, y). g (x, y) * indicator (- RF\<^sup>\<circ>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f)) x * indicator (- RF\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f)) y)"
iT_Div_mem_iff_Int: fixes k :: "nat"   and x :: "nat"   and I :: "nat set" assumes "0 < k" shows "(x \<in> I \<oslash> k) = (I \<inter> [x * k\<dots>,k - Suc 0] \<noteq> {})"
rtrancl_path_nth: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list"   and y :: "'a"   and i :: "nat" assumes "rtrancl_path R x xs y"   and "i < length xs" shows "R ((x # xs) ! i) (xs ! i)"
has_fps_expansion_imp_holomorphic: fixes f :: "complex \<Rightarrow> complex"   and F :: "complex fps" assumes "f has_fps_expansion F" obtains s :: "complex set" where "open s" and "0 \<in> s" and "f holomorphic_on s" and "\<And>z. z \<in> s \<Longrightarrow> f z = eval_fps F z"
L_in_iff: fixes l :: "'lbl"   and A :: "('lbl strand_label \<times> ('fun, ('fun, 'atom) Term.term \<times> nat) stateful_strand_step) list" shows "(l \<in> set (L A)) = (\<exists>a\<in>set A. is_LabelN l a)"
stationary_distribution_def: fixes N :: "'a pmf" shows "K1.stationary_distribution N = (N = N \<bind> K1)"
times_monomial_left: fixes t :: "'c"   and c :: "'d"   and p :: "'c \<Rightarrow>\<^sub>0 'd" shows "monomial c t * p = MPoly_Type_Class.punit.monom_mult c t p"
assms_2: shows "\<BB> \<subseteq>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"
run_rel_def: fixes R :: "('c \<times> 'd) set" shows "run_rel R \<equiv> {(ra, rb). \<forall>i. (ra i, rb i) \<in> R}"
Dom_memb: fixes f :: "'c local.term set"   and t :: "'c local.term" assumes "\<F>C.arr f"   and "t \<in> f" shows "Dom t = \<F>C.DOM f"
above: fixes x :: "'a" assumes "x \<in> A" shows "x \<sqsubseteq> s"
set_to_sorted_list_def: fixes R :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and S :: "'f set" shows "set_to_sorted_list R S \<equiv> it_to_sorted_list R S"
q: shows "p = [:- a, 1::'a:] ^ order a p * q"
cos_double_cos: fixes w :: "'a" shows "cos ((2::'a) * w) = (2::'a) * (cos w)\<^sup>2 - (1::'a)"
SeqAppendP_NotInDom: fixes f1 :: "tm"   and k1 :: "tm"   and f2 :: "tm"   and k2 :: "tm"   and g :: "tm"   and k :: "tm" shows "{SeqAppendP f1 k1 f2 k2 g, HaddP k1 k2 k, OrdP k1} \<turnstile> NotInDom k g"
restrict_space_def: fixes M :: "'a measure"   and \<Omega> :: "'a set" shows "restrict_space M \<Omega> = measure_of (\<Omega> \<inter> space M) (sets.restricted_space M \<Omega>) (emeasure M)"
real_less_eq_code: fixes x :: "rat"   and y :: "rat" shows "(Ratreal x \<le> Ratreal y) = (x \<le> y)"
silent_moves_intra_path: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and m :: "'node"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as :: "'edge list"   and m' :: "'node"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "S,f \<turnstile> (m # ms,s) =as\<Rightarrow>\<^sub>\<tau> (m' # ms',s')"   and "\<forall>a\<in>set as. intra_kind (kind a)" shows "ms = ms' \<and> get_proc m = get_proc m'"
f_finite: shows "f (source \<Delta>, x) \<noteq> \<top>"
cases: fixes x :: "'a list" obtains "x = []"   | xa :: "'a" and l' :: "'a list" where "x = xa # l'"
B_inverse: fixes y :: "'c" assumes "y \<in> UNIV" shows "un_B (B y) = y"
bound_intersect_2d_ud_segments_of_aform: fixes prec :: "nat"   and X :: "(real \<times> real) \<times> (real \<times> real) pdevs"   and xl :: "real"   and m0 :: "real"   and M0 :: "real"   and e :: "nat \<Rightarrow> real" assumes "bound_intersect_2d_ud prec X xl = Some (m0, M0)"   and "e \<in> UNIV \<rightarrow> {- 1..1}" shows "{aform_val e X} \<inter> {x. fst x = xl} \<subseteq> {(xl, m0)..(xl, M0)}"
II: fixes it :: "'x set"   and \<sigma> :: "'a" assumes "it \<subseteq> \<alpha> s"   and "it \<noteq> {}"   and "\<not> c \<sigma>"   and "I it \<sigma>"   and "\<forall>k\<in>it. \<forall>j\<in>\<alpha> s - it. j \<le> k" shows "P \<sigma>"
drop_take_1: fixes n :: "nat"   and xs :: "'d list" assumes "n < length xs" shows "xs \<up> n \<down> Suc 0 = [xs ! n]"
subcategory_axioms: shows "subcategory Span.VxVxV.comp (\<lambda>\<tau>\<mu>\<nu>. Span.arr (fst \<tau>\<mu>\<nu>) \<and> Span.VV.arr (snd \<tau>\<mu>\<nu>) \<and> Span.src (fst \<tau>\<mu>\<nu>) = Span.trg (fst (snd \<tau>\<mu>\<nu>)))"
fdia_d_simp: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set" shows "relfdia x (rdom y) = relfdia x y"
comp_reduce: fixes f :: "'d"   and g :: "'d"   and k :: "'d"   and h :: "'d" assumes "f \<cdot>\<^sub>D g = k"   and "D.seq f g"   and "D.seq g h" shows "f \<cdot>\<^sub>D g \<cdot>\<^sub>D h = k \<cdot>\<^sub>D h"
Cauchy_derivative_integral_circlepath_1: fixes z :: "complex"   and r :: "real"   and f :: "complex \<Rightarrow> complex"   and w :: "complex" assumes "continuous_on (cball z r) f"   and "f holomorphic_on ball z r"   and "w \<in> ball z r" shows "(\<lambda>u. f u / (u - w)\<^sup>2) contour_integrable_on circlepath z r"
stc_add: fixes x :: "complex star"   and y :: "complex star" assumes "x \<in> HFinite"   and "y \<in> HFinite" shows "stc (x + y) = stc x + stc y"
unique_norm: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C"   and "free_monoidal_category.ARR C f" shows "\<exists>!t. \<forall>u. u \<in> f \<longrightarrow> free_monoidal_category.norm C u = t"
mod_in_res_units: fixes a :: "int" assumes "1 < m"   and "coprime a m" shows "a mod m \<in> Units R"
singleton_adjacent_0: fixes S :: "'b set"   and s :: "'b" assumes "CoxeterComplex S"   and "s \<in> S" shows "dual_order.poset_simplex_map (PreCoxeterSystem.special_cosets S) {s} \<sim> dual_order.poset_simplex_map (PreCoxeterSystem.special_cosets S) 0"
assms: shows "#\<^sub>V \<AA> = size_atom bs1_bs2"
extend_measure_caratheodory: fixes M :: "'b measure"   and \<Omega> :: "'b set"   and I :: "'c set"   and G :: "'c \<Rightarrow> 'b set"   and \<mu> :: "'c \<Rightarrow> ennreal"   and i :: "'c" assumes "M = extend_measure \<Omega> I G \<mu>"   and "i \<in> I"   and "semiring_of_sets \<Omega> (G ` I)"   and "\<And>i. \<lbrakk>i \<in> I; G i = {}\<rbrakk> \<Longrightarrow> \<mu> i = 0"   and "\<And>i j. \<lbrakk>i \<in> I; j \<in> I; G i = G j\<rbrakk> \<Longrightarrow> \<mu> i = \<mu> j"   and "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> \<mu> i"   and "\<And>A j. \<lbrakk>A \<in> UNIV \<rightarrow> I; j \<in> I; disjoint_family (G \<circ> A); (\<Union>i. G (A i)) = G j\<rbrakk> \<Longrightarrow> (\<Sum>n. \<mu> (A n)) = \<mu> j" shows "emeasure M (G i) = \<mu> i"
sconf: shows "P,E \<turnstile> s\<^sub>0 \<surd>"
70: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> x \<sqinter> - - y = x"
odd_bit_iff_bit_pred: fixes a :: "'a"   and n :: "nat" assumes "odd a" shows "bit a n = (bit (a - (1::'a)) n \<or> n = 0)"
plus_gt_right_mono: fixes default :: "'c"   and gt :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and a :: "'c" assumes "one_mono_ordered_semiring_1 default gt"   and "gt x y" shows "gt (a + x) (a + y)"
lm018: fixes G :: "'c set"   and N :: "'d set" shows "possible_allocations_rel G N = injectionsUniverse \<inter> {a. Domain a \<in> all_partitions G \<and> Range a \<subseteq> N}"
fmdom'_add: fixes m :: "('c, 'd) fmap"   and n :: "('c, 'd) fmap" shows "fmdom' (m ++\<^sub>f n) = fmdom' m \<union> fmdom' n"
simps_4: fixes f :: "'d \<Rightarrow> 'c"   and p :: "('b, 'd) fm"   and q :: "('b, 'd) fm" shows "sub f (p \<^bold>\<or> q) = (sub f p \<^bold>\<or> sub f q)"
is_leq: fixes a :: "rat"   and n :: "QDelta ns_constraint" assumes "(a, n) \<in> set N" shows "n \<in> set (concat (map constraint_to_qdelta_constraint csl)) \<and> is_leq_ns (a *R n) \<and> a \<noteq> 0"
frees_list_comb: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and abs_pred :: "('b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "'b"   and xs :: "'b list" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "frees (term.list_comb app t xs) = frees t |\<union>| pre_term.freess frees xs"
eq: fixes x :: "'a" assumes "x \<in> A"   and "x \<noteq> a" shows "f x = g x"
well_def_find_zeros_from_lhs_vec_R: fixes subsets :: "(nat list \<times> nat list) list"   and signs :: "rat list list"   and p :: "real poly"   and qs :: "real poly list"   and j :: "nat" assumes "length subsets = length signs"   and "invertible_mat (M_mat_R signs subsets)"   and "p \<noteq> 0"   and "well_def_signs (length qs) signs"   and "distinct signs"   and "set (characterize_consistent_signs_at_roots p qs) \<subseteq> set signs"   and "satisfy_equation_R p qs subsets signs"   and "j \<in> set (find_nonzeros_from_input_vec (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))" shows "j < length (cols (M_mat_R signs subsets))"
self_in_upset: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.order less_eq less" shows "x \<in> {y. less_eq x y}"
rec: fixes m :: "'node" assumes "v \<notin> defs g n"   and "predecessors g n = [m]"   and "m \<in> set (\<alpha>n g)"   and "lookupDef g n v = lookupDef g m v" shows "thesis"
topological_basis: fixes "open" :: "'b set \<Rightarrow> bool"   and B :: "'b set set" assumes "class.topological_space open" shows "topological_space.topological_basis open B = (\<forall>x. open x = (\<exists>B'\<subseteq>B. \<Union> B' = x))"
Cons_sorted_less_nth: fixes xs :: "'a list"   and p :: "'a" assumes "\<forall>x<length xs. xs ! x < p"   and "Sorted_Less.sorted (rev xs)" shows "Sorted_Less.sorted (rev (p # xs))"
awp_iff_stable: fixes A :: "'b set"   and F :: "'b program" shows "(A \<subseteq> awp F A) = (F \<in> stable A)"
gc_W_empty_invL_def: shows "gc_W_empty_invL = (\<lambda>s. (atS gc get_roots_UN_get_work_locs s \<longrightarrow> (\<forall>x. mut_m.gc_W_empty_mut_inv x s\<down>)) \<and> (at gc mark_loop_get_roots_load_W s \<longrightarrow> sys_W s\<down> = {} \<longrightarrow> no_grey_refs s\<down>) \<and> (at gc mark_loop_get_work_load_W s \<longrightarrow> sys_W s\<down> = {} \<longrightarrow> no_grey_refs s\<down>) \<and> (at gc mark_loop s \<longrightarrow> gc_W s\<down> = {} \<longrightarrow> no_grey_refs s\<down>) \<and> (atS gc no_grey_refs_locs s \<longrightarrow> no_grey_refs s\<down>) \<and> (atS gc gc_W_empty_locs s \<longrightarrow> gc_W s\<down> = {}))"
div_le_mod_le_imp_le: fixes a :: "nat"   and m :: "nat"   and b :: "nat" assumes "a div m \<le> b div m"   and "a mod m \<le> b mod m" shows "a \<le> b"
SeqCTerm_imp_wf_dbtm: fixes vf :: "bool"   and s :: "hf"   and k :: "hf"   and x :: "hf"   and e :: "(name, hf) finfun" assumes "SeqCTerm vf s k x" shows "\<exists>t. wf_dbtm t \<and> x = \<lbrakk>quot_dbtm t\<rbrakk>e"
well_formed_from_rel: fixes rel :: "('a \<times> ('b \<times> 'c) \<times> 'a) set"   and q0 :: "'a" assumes "ASC_Example.well_formed_rel rel" shows "well_formed (from_rel rel q0)"
pdevs_val_sum: fixes e :: "nat \<Rightarrow> real"   and X :: "'a pdevs" shows "pdevs_val e X = (\<Sum>i<degree X. e i *\<^sub>R pdevs_apply X i)"
Restrp_well_founded: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "well_founded A less" shows "well_founded UNIV (less \<restriction> A)"
convex_hull_eq: fixes s :: "'a set" shows "(convex hull s = s) = convex s"
rtrancl_of_preorder: fixes Rel :: "('a \<times> 'a) set" assumes "preorder Rel" shows "Rel\<^sup>* = Rel"
mult_mat_vec_scalar: fixes A :: "(('d, 'f) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec"   and v :: "('d, 'f) Finite_Cartesian_Product.vec" shows "A *v v = HMA_Connect.mk_vec (\<lambda>i. scalar_product (row i A) v)"
assms_4: shows "rel_set T xl (a' \<union> b')"
hd_w_suc_lt_n: shows "\<forall>q\<in>succs (hd w). q < n"
simps_4: fixes m1 :: "'b match_expr"   and m2 :: "'b match_expr" shows "has_primitive (MatchAnd m1 m2) = (has_primitive m1 \<or> has_primitive m2)"
weakenBisimWeakCoinductAux: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and X :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "(\<Psi>, P, Q) \<in> X"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> \<Psi> \<rhd> P \<lessapprox>\<^sub>w<X> Q \<and> \<Psi> \<rhd> P \<leadsto>\<^sub>w<X> Q \<and> (\<forall>\<Psi>'. (\<Psi> \<otimes> \<Psi>', P, Q) \<in> X) \<and> (\<Psi>, Q, P) \<in> X" shows "\<Psi> \<rhd> P \<approx>\<^sub>w Q"
proj_pick: fixes P :: "'a \<Rightarrow> bool"   and \<phi> :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and X :: "'a set" assumes "equivP P \<phi>"   and "(P /// \<phi>) X" shows "proj \<phi> (pick X) = X"
verit_eq_simplify_10: fixes x2 :: "num" shows "num.One \<noteq> num.Bit0 x2"
that: fixes \<pi>' :: "nat \<Rightarrow> 'n"   and n' :: "nat" assumes "is_path \<pi>'"   and "\<pi>' 0 = x"   and "\<pi>' n' = local.return"   and "\<forall>i>0. \<pi>' i \<noteq> x" shows "thesis"
0: shows "exu out (instInp \<tau> t) = subst (exu out \<tau>) t inp"
mult_botr: fixes x :: "'a" shows "x \<cdot> \<bottom> = \<bottom>"
encF: fixes f :: "'trm \<Rightarrow> 'trm" assumes "f \<in> F" shows "encF f \<in> ptrm (Suc 0)"
surj_imp_inv_eq: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'c" assumes "surj f"   and "\<And>x. g (f x) = x" shows "inv f = g"
diff_single_eq_union: fixes x :: "'a"   and M :: "'a multiset"   and N :: "'a multiset" assumes "x \<in># M" shows "(M - {#x#} = N) = (M = add_mset x N)"
dist_triangle_half_r: fixes y :: "'b"   and x1 :: "'b"   and e :: "real"   and x2 :: "'b" assumes "dist y x1 < e / 2"   and "dist y x2 < e / 2" shows "dist x1 x2 < e"
of_int_eq_1_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and z :: "int" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(ring_1.of_int one plus zero minus z = one) = (z = 1)"
dim_eliminate_entries_i2_1: fixes z :: "'b"   and mm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and tt :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and v :: "integer \<Rightarrow> 'b"   and m :: "'b mat_impl"   and i :: "integer" shows "dim_row_impl (eliminate_entries_i2 z mm tt v m i) = dim_row_impl m"
divide_inverse_commute: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.field divide inverse times one plus zero minus uminus" shows "divide a b = times (inverse b) a"
lex_ext_least_1: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and m :: "nat"   and xs :: "'a list" shows "snd (lex_ext f m xs [])"
mem6: shows "rotate (n - 1) ls = rotate (m - 1 + |ms|) ms"
var_self: fixes v :: "'b" shows "[(v, Var v)] \<doteq> []"
simps_3: fixes p :: "int \<times> int"   and p\<^sub>0 :: "int \<times> int"   and v :: "int \<times> int"   and va :: "(int \<times> int) list" shows "find_closest_bf p (p\<^sub>0 # v # va) = (let p\<^sub>1 = find_closest_bf p (v # va) in if dist (case p of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) < dist (case p of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) then p\<^sub>0 else p\<^sub>1)"
induct: fixes x1 :: "(String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and x2 :: "typ"   and x3 :: "String.literal set"   and P :: "(String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option) \<Rightarrow> typ \<Rightarrow> String.literal set \<Rightarrow> bool" assumes "has_sort x1 x2 x3"   and "\<And>sub S S' tcs a. sort_leq sub S S' \<Longrightarrow> P (sub, tcs) (Tv a S) S'"   and "\<And>tcs \<kappa> dm S sub Ts. \<lbrakk>tcs \<kappa> = Some dm; \<forall>c\<in>S. \<exists>Ss. dm c = Some Ss \<and> list_all2 (\<lambda>x1 x2. has_sort (sub, tcs) x1 x2 \<and> P (sub, tcs) x1 x2) Ts Ss\<rbrakk> \<Longrightarrow> P (sub, tcs) (Ty \<kappa> Ts) S" shows "P x1 x2 x3"
p_mono: fixes x :: "real"   and y :: "real" assumes "x \<le> y"   and "y \<le> delayed_safe_distance.other.t_stop" shows "delayed_safe_distance.other.p x \<le> delayed_safe_distance.other.p y"
learn_finE_1: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option" assumes "learn_fin \<psi> U s" shows "environment \<psi> U s"
iter_narrow_step_ivl_termination: fixes c0 :: "ivl Abs_State.st option acom" assumes "c0 \<in> Com (vars (strip c0))"   and "step_ivl \<top> c0 \<sqsubseteq> c0" shows "\<exists>c. iter_narrow (step_ivl \<top>) c0 = Some c"
hyps_3: fixes x :: "'a" assumes "x \<in> M" shows "x \<in> iterates_above a"
has_fields_types: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and FDTs :: "((char list \<times> char list) \<times> ty) list"   and FD :: "char list \<times> char list"   and T :: "ty"   and wf_md :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'b \<Rightarrow> bool" assumes "P \<turnstile> C has_fields FDTs"   and "(FD, T) \<in> set FDTs"   and "wf_prog wf_md P" shows "is_type P T"
assms_2: shows "m \<le>\<^sub>n SPEC \<Psi>"
countable_finite_comprehension: fixes P :: "'d set \<Rightarrow> bool"   and f :: "'d set \<Rightarrow> 'e set"   and M :: "'e measure" assumes "\<And>s. P s \<Longrightarrow> finite s"   and "\<And>s. P s \<Longrightarrow> f s \<in> sets M" shows "\<Union> {f s |s. P s} \<in> sets M"
NextPropagate'_causes_safe_1: fixes c0 :: "('b, 't, 'loc) Combined.configuration"   and c1 :: "('b, 't, 'loc) Combined.configuration"   and p :: "'b" assumes "NextPropagate' c0 c1 p"   and "inv_imps_work_sum (prop_config c1 p)"   and "inv_implications_nonneg (prop_config c1 p)" shows "safe (prop_config c1 p)"
to_mregex_exec_ok: fixes r :: "Formula.formula Regex.regex"   and xs :: "Formula.formula list"   and mr :: "mregex"   and ys :: "Formula.formula list" assumes "to_mregex_exec r xs = (mr, ys)" shows "\<exists>zs. ys = xs @ zs \<and> set zs = atms r \<and> ok (length ys) mr"
hyps_2: fixes X :: "tyvrs"   and T\<^sub>2 :: "ty"   and T\<^sub>1 :: "ty" assumes "S = (\<forall>X<:T\<^sub>1. T\<^sub>2)"   and "val t" shows "\<exists>X ta S. t = (\<lambda>X<:S. ta)"