hcju/isabelleps huggingface dataset:
has_fields_mono_lem: fixes D :: "char list"   and C :: "char list"   and P :: "'b prog"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list" assumes "P \<turnstile> D \<preceq>\<^sup>* C"   and "P \<turnstile> C has_fields FDTs" shows "\<exists>pre. P \<turnstile> D has_fields pre @ FDTs \<and> dom (map_of pre) \<inter> dom (map_of FDTs) = {}"
absolutely_continuous_def: fixes M :: "'b measure"   and N :: "'b measure" shows "absolutely_continuous M N = (null_sets M \<subseteq> null_sets N)"
heap_free_def: fixes n :: "nat" shows "heap_free n = (\<lambda>s. atleast_free (heap (snd s)) n)"
set_times_intro2: fixes b :: "'b"   and C :: "'b set"   and a :: "'b" assumes "b \<in> C" shows "a * b \<in> a *o C"
add_assoc: fixes a :: "('a \<times> 'a) set"   and b :: "('a \<times> 'a) set"   and c :: "('a \<times> 'a) set" assumes "valid_frac a"   and "valid_frac b"   and "valid_frac c" shows "add_rel (add_rel a b) c = add_rel a (add_rel b c)"
intros_2: fixes c :: "('loc, 't) configuration"   and loc1 :: "'loc"   and loc2 :: "'loc"   and M :: "'t zmultiset"   and xs :: "('loc \<times> 'sum \<times> 'loc) list"   and s :: "'sum"   and loc3 :: "'loc" assumes "loc_imps_fw c loc1 loc2 M xs"   and "s \<in>\<^sub>A summary loc2 loc3"   and "distinct (xs @ [(loc2, s, loc3)])" shows "loc_imps_fw c loc1 loc3 ({#results_in t s. t \<in>#\<^sub>z M#} + c_imp c loc3) (xs @ [(loc2, s, loc3)])"
e1\<alpha>1_C_empty: shows "(e1 # \<alpha>1) \<upharpoonleft> C\<^bsub>\<V>\<^esub> = []"
is_nnf_nnf: fixes F :: "'b formula" shows "is_nnf (nnf F)"
component_edges_subset: fixes g :: "'b ugraph"   and r :: "'b" shows "edges (component_of g r) \<subseteq> edges g"
collect_pseudominimals_below_in_less_eq_top: fixes P :: "'a set"   and x :: "'a"   and f :: "'a \<Rightarrow> 'b set"   and A :: "'b set"   and a :: "'a set" assumes "OrderingSetIso (\<^bold>\<le>) (\<^bold><) dual_order.greater_eq dual_order.greater (P .\<^bold>\<le> x) f"   and "f ` (P .\<^bold>\<le> x) = Pow A"   and "order.greater_eq {y. pseudominimal_in (P .\<^bold>\<le> x) y} a" shows "x \<succeq> the_inv_into (P .\<^bold>\<le> x) f (\<Union> (f ` a))"
quantale_closed_Sup_maps_def: shows "quantale_closed_Sup_maps = {f. (\<forall>x y. f x \<cdot> f y \<le> f (x \<cdot> y)) \<and> Sup_pres f}"
prems_2: shows "is' \<lhd> dims (prod_list (B # Bs))"
1: shows "\<pp>\<^sub>1[a, b] \<cdot> \<langle>f, g\<rangle> = f \<and> \<pp>\<^sub>0[a, b] \<cdot> \<langle>f, g\<rangle> = g"
listrel1_rtrancl_subset_rtrancl_listrel1: fixes r :: "('g \<times> 'g) set" shows "listrel1 (r\<^sup>*) \<subseteq> (listrel1 r)\<^sup>*"
ord_Zp_ultrametric_eq: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> nonzero Zp"   and "y \<in> nonzero Zp"   and "ord_Zp y < ord_Zp x" shows "ord_Zp (x \<oplus> y) = ord_Zp y"
distinct_RS2: fixes p :: "('b, 'c) Combinators list" assumes "distinct p" shows "distinct (removeShadowRules2 p)"
has_integral_improper_at_topI: fixes f :: "real \<Rightarrow> 'a"   and I :: "real \<Rightarrow> 'a"   and a :: "real"   and J :: "'a" assumes "\<forall>\<^sub>F k in at_top. (f has_integral I k) {a..k}"   and "(I \<longlongrightarrow> J) at_top" shows "(f has_integral J) {a..}"
vappend_def: fixes xs :: "V"   and ys :: "V" shows "xs @\<^sub>\<circ> ys = vfsequence_of_vlist (vlist_of_vfsequence ys @ vlist_of_vfsequence xs)"
steps_append': fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and as :: "'c list"   and bs :: "'c list"   and xs :: "'c list" assumes "Graph_Defs.steps E as"   and "Graph_Defs.steps E bs"   and "last as = hd bs"   and "as @ tl bs = xs" shows "Graph_Defs.steps E xs"
matrix_vector_mul_diag_mat: fixes f :: "'c \<Rightarrow> 'b"   and s :: "('b, 'c) vec" shows "diag_mat f *v s = (\<chi>i. f i * s $ i)"
simps_15: fixes x1 :: "'d" shows "funs_term (Var x1) = {}"
add_less_same_cancel1: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and b :: "'b"   and a :: "'b" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less (plus b a) b = less a zero"
img_is_subfield_2: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e"   and K :: "'c set" assumes "ring_hom_ring R S h"   and "subfield K R"   and "\<one>\<^bsub>S\<^esub> \<noteq> \<zero>\<^bsub>S\<^esub>" shows "subfield (h ` K) S"
mset_pos_uminus: fixes A :: "'b zmultiset" shows "mset_pos (- A) = mset_neg A"
square_free_m_smultD: fixes c :: "int"   and f :: "int poly" assumes "q.square_free_m (Polynomial.smult c f)" shows "q.square_free_m f"
subsetCE: fixes A :: "'a set"   and B :: "'a set"   and c :: "'a" assumes "A \<subseteq> B" obtains "c \<notin> A"   | "c \<in> B"
swap_def: fixes p :: "'c \<times> 'b" shows "prod.swap p = (snd p, fst p)"
run_get_nondet: fixes get :: "('f \<Rightarrow> 'd) \<Rightarrow> 'd"   and f :: "'f \<Rightarrow> ('e, 'd) nondetT" shows "run_nondet (get_nondet get f) = get (\<lambda>s. run_nondet (f s))"
suffixI: fixes ys :: "'b list"   and zs :: "'b list"   and xs :: "'b list" assumes "ys = zs @ xs" shows "suffix xs ys"
image_mset_thm: fixes E :: "'b multiset"   and f :: "'c \<Rightarrow> 'b"   and E' :: "'c multiset"   and x :: "'b" assumes "E = image_mset f E'"   and "x \<in># E" shows "\<exists>y. y \<in># E' \<and> x = f y"
less_multiset\<^sub>H\<^sub>O_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and M :: "'b multiset"   and N :: "'b multiset" assumes "class.preorder less_eq less" shows "preorder.less_multiset\<^sub>H\<^sub>O less M N = (M \<noteq> N \<and> (\<forall>y. count N y < count M y \<longrightarrow> (\<exists>x. less y x \<and> count M x < count N x)))"
nres_order_simps_4: fixes M :: "'b nres" shows "(FAIL \<le> M) = (M = FAIL)"
is_path'_conc: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> b"   and j :: "'a"   and xs :: "'a list"   and i :: "'a"   and ys :: "'a list"   and k :: "'a" assumes "is_path' r j xs i"   and "is_path' r i ys k" shows "is_path' r j (xs @ i # ys) k"
prems: shows "wt t \<ge>\<^sub>p wt (App s1 s2)"
PPs_closed_except': fixes t :: "'c \<Rightarrow>\<^sub>0 nat"   and X :: "'c set"   and Y :: "'c set" assumes "t \<in> .[X]" shows "except t Y \<in> .[X - Y]"
card_subset_eq: fixes B :: "'d set"   and A :: "'d set" assumes "finite B"   and "A \<subseteq> B"   and "card A = card B" shows "A = B"
these_image_Some_eq: fixes A :: "'e set" shows "Option.these (Some ` A) = A"
a_star_AndL1: fixes M :: "trm"   and M' :: "trm"   and x :: "name"   and y :: "name" assumes "M \<longrightarrow>\<^sub>a* M'" shows "AndL1 x.M y \<longrightarrow>\<^sub>a* AndL1 x.M' y"
lenlex_transI: fixes r :: "('a \<times> 'a) set" assumes "trans r" shows "trans (lenlex r)"
card_of_singl_ordLeq: fixes A :: "'a set"   and b :: "'b" assumes "A \<noteq> {}" shows "|{b}| \<le>o |A|"
assoc_simps_2: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "B.ide f"   and "B.ide g"   and "B.ide h"   and "src\<^sub>B f = trg\<^sub>B g"   and "src\<^sub>B g = trg\<^sub>B h" shows "src\<^sub>B \<a>\<^sub>B[f, g, h] = src\<^sub>B h"
gnodes_nodes: fixes A :: "('b, 'a) nba"   and w :: "'b stream"   and V :: "(nat \<times> 'a) set" shows "gnodes A w V \<subseteq> UNIV \<times> nodes A"
x0_3: fixes x :: "real" assumes "x0 \<le> x" shows "(g has_real_derivative g' x) (at x)"
equiv_subst_var: fixes subst_var :: "nat \<Rightarrow> linear_poly \<Rightarrow> linear_poly \<Rightarrow> linear_poly"   and v :: "nat \<Rightarrow> 'a"   and x\<^sub>j :: "nat"   and lp' :: "linear_poly"   and lp :: "linear_poly" assumes "SubstVar TYPE('a) subst_var" shows "v x\<^sub>j = lp' \<lbrace> v \<rbrace> \<longrightarrow> lp \<lbrace> v \<rbrace> = subst_var x\<^sub>j lp' lp \<lbrace> v \<rbrace>"
mult_left_less_imp_less: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e"   and a :: "'e"   and b :: "'e" assumes "class.linordered_semiring plus minus less_eq less zero times"   and "less (times c a) (times c b)"   and "less_eq zero c" shows "less a b"
R_skip: fixes p :: "'a" shows "(1::'a) \<le> Ref p p"
negligible_hyperplane: fixes a :: "'c"   and b :: "real" assumes "a \<noteq> (0::'c) \<or> b \<noteq> 0" shows "negligible {x. a \<bullet> x = b}"
heqr: fixes x :: "'a" assumes "x \<in> W \<union> T" shows "h x = r x"
neqE: fixes x :: "'b word"   and y :: "'b word" assumes "x \<noteq> y" obtains "x <s y"   | "y <s x"
C_eq_id: fixes p :: "((int \<times> int) set set, int) Combinators list" assumes "wellformed_policy1_strong p" shows "C (list2FWpolicy (insertDeny p)) = C (list2FWpolicy p)"
psubsetD: fixes A :: "'a set"   and B :: "'a set"   and c :: "'a" assumes "A \<subset> B"   and "c \<in> A" shows "c \<in> B"
cd_is_cs_less: fixes l :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat" assumes "l cd\<^bsup>\<pi>\<^esup>\<rightarrow> k" shows "cs\<^bsup>\<pi>\<^esup> k \<prec> cs\<^bsup>\<pi>\<^esup> l"
N_d: shows "d (Suc j) \<le> N ^ Suc j"
next_dom_events_def: fixes P :: "'c process"   and D :: "'c \<Rightarrow> 'd"   and u :: "'d"   and xs :: "'c list" shows "next_dom_events P D u xs \<equiv> {x. u = D x \<and> x \<in> next_events P xs}"
wadjust_loop_on_left_moving_snd_nonempty_3: fixes m :: "nat"   and rs :: "nat" shows "wadjust_erase2 m rs ([], []) = False"
countable_image_inj_eq: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "inj_on f S" shows "countable (f ` S) = countable S"
ass_upd_same: fixes v :: "'b"   and u :: "'a"   and a :: "('b \<times> 'a) list"   and ass :: "'b \<Rightarrow> 'a" shows "update_assignment ((v, u) # a) ass v = u"
REL: shows "(Gi, G) \<in> bg_impl_rel_ext unit_rel Id"
S_def: shows "S \<equiv> if card Q = 2 then {segment a b} else {segment (f i) (f (i + 1)) |i. i < card Q - 1}"
path_C_to_C_unique: fixes wf_md :: "(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 \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool"   and 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 C :: "char list" assumes "wf_prog wf_md P"   and "is_class P C" shows "P \<turnstile> Path C to C unique"
get_scdom_component_def: shows "get_scdom_component \<equiv> l_get_scdom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_scdom_component get_owner_document get_disconnected_nodes to_tree_order"
assms_3: shows "triangle_circumscribes_triangle A' B' C' A B C"
right_adjoint_simps_2: fixes f :: "'a" assumes "C.is_left_adjoint f" shows "src\<^sub>C f\<^sup>*\<^sup>C = trg\<^sub>C f"
dependent_mono: fixes B :: "('a, 'b) vec set"   and A :: "('a, 'b) vec set" assumes "vec.dependent B"   and "B \<subseteq> A" shows "vec.dependent A"
funcsetI: fixes A :: "'g set"   and f :: "'g \<Rightarrow> 'h"   and B :: "'h set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B" shows "f \<in> A \<rightarrow> B"
mset_set_set_mset_subseteq: fixes M :: "'a multiset" shows "mset_set (set_mset M) \<subseteq># M"
prod_cases: fixes P :: "'d \<times> 'e \<Rightarrow> bool"   and p :: "'d \<times> 'e" assumes "\<And>a b. P (a, b)" shows "P p"
ide_Ide: fixes C1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'a \<times> 'b" assumes "product_category C1 C2"   and "partial_magma.ide C1 (fst a) \<and> partial_magma.ide C2 (snd a)" shows "partial_magma.ide (product_category.comp C1 C2) a"
set_remove1_subset: fixes x :: "'b"   and xs :: "'b list" shows "set (remove1 x xs) \<subseteq> set xs"
transpose_rank_mul_conjugate_transpose: fixes A :: "'b mat"   and n :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat n nc" shows "vec_space.rank nc A\<^sup>H \<le> vec_space.rank n (A * A\<^sup>H)"
length_minus_poly_rev_list: fixes xs :: "'b list"   and ys :: "'b list" shows "length (minus_poly_rev_list xs ys) = length xs"
acrs: shows "all_chains (no_call_to c) \<Gamma> rs"
is_b_graph: fixes m :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'b" shows "b_graph (degeneralize_ext m)"
evret1: shows "eventually (returns_to p1) (at x0 within S)"
mono_def: fixes f :: "'c \<Rightarrow> 'd" shows "mono f = (\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y)"
powser_sums_zero: fixes a :: "nat \<Rightarrow> 'b" shows "(\<lambda>n. a n * (0::'b) ^ n) sums a 0"
degree_sum: fixes n :: "nat"   and f :: "nat \<Rightarrow> real mpoly"   and var :: "nat"   and x :: "nat" assumes "\<forall>i\<in>{0..n}. MPoly_Type.degree (f i) var \<le> x" shows "MPoly_Type.degree (sum f {0..n}) var \<le> x"
S\<^sub>1_def: shows "THfgh.S\<^sub>1 = \<lbrakk>\<lbrakk>tab\<^sub>1 h\<rbrakk>\<rbrakk>"
inorder_del: fixes t :: "'a bro"   and h :: "nat"   and x :: "'a" assumes "t \<in> T h"   and "sorted (inorder t)" shows "inorder (del x t) = del_list x (inorder t)"
projection_eqI: fixes M :: "'a set"   and h :: "'a"   and x :: "'a" assumes "closed_csubspace M"   and "h - x \<in> orthogonal_complement M"   and "x \<in> M" shows "projection M h = x"
epair_None: fixes a :: "'a"   and b :: "'a" shows "(epair {a, b} = None) = (a = b)"
viterbi_viterbi_prob: fixes s :: "'s"   and t_end :: "'s"   and os :: "'t list" shows "snd (viterbi s t_end os) = viterbi_prob s t_end os"
single_valued_below_Id: fixes R :: "('a \<times> 'a) set" assumes "R \<subseteq> Id" shows "single_valued R"
distinct_1: fixes x11 :: "'d"   and x12 :: "'d"   and x13 :: "('d, 'e, 'f) psi"   and x21 :: "'d"   and x22 :: "('d, 'e, 'f) boundOutput" shows "RIn_Rep x11 x12 x13 \<noteq> ROut_Rep x21 x22"
uint32_test_bit_def: fixes x :: "uint32"   and n :: "integer" shows "uint32_test_bit x n = (if n < 0 \<or> 31 < n then undefined bit x n else bit x (nat_of_integer n))"
closedin_subset: fixes U :: "'b topology"   and S :: "'b set" assumes "closedin U S" shows "S \<subseteq> topspace U"
coprime: shows "comm_monoid_mult_class.coprime (#v) (#w)"
norm_le_l1: fixes x :: "'c" shows "norm x \<le> (\<Sum>b\<in>Basis. \<bar>x \<bullet> b\<bar>)"
deg_const: fixes a :: "int" assumes "a \<in> carrier R" shows "deg R (monom (UP R) a 0) = 0"
nextVert: shows "f \<bullet> v = u \<longrightarrow> n \<noteq> 0"
1: fixes g :: "'t \<Rightarrow>\<^sub>0 'b" shows "q' g \<in> punit.dgrad_p_set d m"
yes_nb: shows "bound0 (list_conj (fst (partition bound0 (conjuncts p))))"
steps_reaches1: fixes x :: "'a"   and xs :: "'a list"   and y :: "'a" assumes "steps (x # xs @ [y])" shows "x \<rightarrow>\<^sup>+ y"
simps: fixes le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b list" shows "linorder_list0.sortedP le a = (a = [] \<or> (\<exists>ys x. a = x # ys \<and> (\<forall>y\<in>set ys. le x y) \<and> linorder_list0.sortedP le ys))"
roundup: fixes x :: "nat"   and y :: "nat" shows "roundup x y = (if x mod y = 0 then x div y else x div y + 1)"
le_fin_le_fininf_transp: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a list"   and w\<^sub>3 :: "'a stream" assumes "traces ind"   and "traces.le_fin ind w\<^sub>1 w\<^sub>2"   and "traces.le_fininf ind w\<^sub>2 w\<^sub>3" shows "traces.le_fininf ind w\<^sub>1 w\<^sub>3"
trace_decomp_head: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and t :: "('a, 'b, 'c) event list"   and c' :: "('a, 'b, 'c) configuration" assumes "trace c (ev # t) c'" shows "\<exists>c''. c \<turnstile> ev \<mapsto> c'' \<and> trace c'' t c'"
One_def: shows "quiver.One \<equiv> MkArr True True []"
foldseq_assoc: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "associative f" shows "foldseq f = foldseq_transposed f"
Unit_Left: fixes hcond :: "'a hrel \<Rightarrow> 'a hrel"   and utp_unit :: "'a hrel"   and P :: "'a hrel" assumes "utp_theory_left_unital hcond utp_unit"   and "P is hcond" shows "utp_unit ;; P = P"
all_security_requirements_fulfilled_imp_get_offending_empty: fixes M :: "'b SecurityInvariant_configured list"   and G :: "'b graph" assumes "valid_reqs M"   and "all_security_requirements_fulfilled M G" shows "get_offending_flows M G = {}"
mult_ac_1: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "a * b * c = a * (b * c)"
filterlim_at_bot_le: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and c :: "'b" shows "filterlim f at_bot F = (\<forall>Z\<le>c. \<forall>\<^sub>F x in F. f x \<le> Z)"
iso_inv_block_in: fixes x :: "'b set" assumes "x \<in># \<B>'" shows "x \<in> (`) \<pi> ` set_mset \<B>"
ordinal_plus_strict_monoR: fixes y :: "ordinal"   and y' :: "ordinal"   and x :: "ordinal" assumes "y < y'" shows "x + y < x + y'"
cod_eval: fixes f :: "'c local.term" assumes "Arr f" shows "cod \<lbrace>f\<rbrace> = \<lbrace>Cod f\<rbrace>"
a1: shows "sets M = sets N \<and> Filtration.filtration M F \<and> (\<forall>t A. A \<in> sets (F t) \<longrightarrow> (emeasure M A = 0) = (emeasure N A = 0))"
is_valid_problem_dom_of_goal_state_is: fixes \<Pi> :: "'b strips_problem" assumes "is_valid_problem_strips \<Pi>" shows "dom (\<Pi>\<^sub>G) \<subseteq> set (\<Pi>\<^sub>\<V>)"
image2_def: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c" shows "BNF_Greatest_Fixpoint.image2 A f g = {(f a, g a) |a. a \<in> A}"
IH: shows "xvec \<sharp>* ([xvec yvec] \<bullet>\<^sub>v p)"
circline_set_positive_y_axis_I: fixes z :: "complex" assumes "is_imag z"   and "0 < Im z" shows "of_complex z \<in> positive_y_axis"
equivalentFormulaeTransitivity: fixes formula1 :: "Literal list list"   and formula2 :: "Literal list list"   and formula3 :: "Literal list list" assumes "equivalentFormulae formula1 formula2"   and "equivalentFormulae formula2 formula3" shows "equivalentFormulae formula1 formula3"
path2_unsnoc: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m"   and "2 \<le> length ns" obtains "g \<turnstile> n-butlast ns\<rightarrow>last (butlast ns)" and "last (butlast ns) \<in> set (old.predecessors g m)"
WT_callee: fixes x :: "'h"   and \<I> :: "('h, 'i) \<I>"   and \<I>' :: "('j, 'k) \<I>"   and s :: "('h, 'i, 'j, 'k) converter" assumes "x \<in> outs_\<I> \<I>"   and "\<I>, \<I>' \<turnstile>\<^sub>C s \<surd>" shows "\<I>' \<turnstile>g run_converter s x \<surd>"
uinvarm_def: fixes v :: "'V"   and wl :: "'V set"   and res :: "'V \<Rightarrow> ('V \<times> 'W \<times> 'V) list option"   and it :: "('W \<times> 'V) set"   and \<sigma> :: "('V \<Rightarrow> 'W infty option) \<times> ('V \<Rightarrow> (('V \<times> 'W \<times> 'V) list \<times> 'W) option)" shows "uinvarm v wl res it \<sigma> \<equiv> uinvar v wl res it (\<alpha>s \<sigma>) \<and> dinvarm \<sigma>"
k_def: shows "local.k = (LEAST i. local.F (Suc i) = 0)"
app_vdomainD: fixes a :: "V"   and r :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" shows "\<exists>b. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r"
psubst_new'_1: fixes c :: "'a"   and t :: "'a FOL_Fitting.term"   and x :: "'a" assumes "new_term c t" shows "psubstt (id(c := x)) t = t"
InitErrorReds: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and Cs :: "char list list"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and C' :: "char list" assumes "sh C = \<lfloor>(sfs, Error)\<rfloor>"   and "P \<turnstile> \<langle>RI (C,THROW NoClassDefFoundError) ; Cs \<leftarrow> e,(h, l, sh),b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>INIT C' (C # Cs,False) \<leftarrow> e,(h, l, sh),b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>"
simps: fixes f :: "int vec list \<times> int vec list"   and mu :: "int iarray iarray"   and d :: "int iarray" shows "fim1_state (f, mu, d) = get_nth_im1 f"
D: shows "D (lim_proc (range Y) [+] S) = D (lim_proc (range (\<lambda>i. Y i [+] S)))"
nat_def: shows "nat \<equiv> map_fun Rep_Integ id (\<lambda>(x, y). x - y)"
faulty1: fixes t :: "lam"   and x :: "name"   and xs :: "name list"   and t' :: "lam" assumes "t \<mapsto> x # xs,t'"   and "\<not> x \<sharp> t'" shows "\<not> x \<sharp> bind xs t'"
AF_ne: fixes k :: "nat"   and i :: "nat" shows "AF k i \<noteq> {}"
sel_rews_1: shows "appF\<cdot>\<bottom> = \<bottom>"
llist_all2_mono: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a llist"   and ys :: "'b llist"   and P' :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "llist_all2 P xs ys"   and "\<And>x y. P x y \<Longrightarrow> P' x y" shows "llist_all2 P' xs ys"
antimono_def: fixes f :: "'b \<Rightarrow> 'c" shows "antimono f = (\<forall>x y. x \<le> y \<longrightarrow> f y \<le> f x)"
cos_mono_le: fixes x :: "real"   and y :: "real" assumes "\<bar>x\<bar> \<le> pi"   and "\<bar>y\<bar> \<le> pi" shows "(cos x \<le> cos y) = (\<bar>y\<bar> \<le> \<bar>x\<bar>)"
enres_lift_def: fixes m :: "'d nres" shows "enres_lift m \<equiv> m \<bind> (\<lambda>x. RETURN (Inr x))"
has_lock_has_lock_acquire_locks: fixes l :: "('a \<times> nat) option"   and T :: "'a"   and t :: "'a"   and n :: "nat" assumes "has_lock l T" shows "has_lock (acquire_locks l t n) T"
le_prod_def: fixes p1 :: "'b \<times> 'c"   and p2 :: "'b \<times> 'c" shows "(p1 \<sqsubseteq> p2) = (fst p1 \<sqsubseteq> fst p2 \<and> snd p1 \<sqsubseteq> snd p2)"
2: shows "(\<pi>' l', \<pi> n) \<in> pdg"
set_alternating_list: fixes n :: "nat"   and s :: "'b"   and t :: "'b" shows "order.greater_eq {s, t} (set (alternating_list n s t))"
dependent_explicit: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s :: "'e set" assumes "module scale" shows "module.dependent scale s = (\<exists>t u. finite t \<and> t \<subseteq> s \<and> (\<Sum>v\<in>t. scale (u v) v) = (0::'e) \<and> (\<exists>v\<in>t. u v \<noteq> (0::'d)))"
integrable_spike_finite: fixes S :: "'a set"   and T :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'b" assumes "finite S"   and "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x"   and "f integrable_on T" shows "g integrable_on T"
i_cut_max_empty_1: fixes I :: "'a set"   and t :: "'a" assumes "finite I"   and "Max I < t" shows "I \<down>\<ge> t = {}"
gr0_imp_self_le_mult2: fixes k :: "nat"   and m :: "nat" assumes "0 < k" shows "m \<le> k * m"
resCases: fixes \<Psi> :: "'b"   and x :: "name"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and Prop :: "'a action \<Rightarrow> ('a, 'b, 'c) psi \<Rightarrow> bool" assumes "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>P \<longmapsto> \<alpha> \<prec> P'"   and "x \<sharp> \<Psi>"   and "x \<sharp> \<alpha>"   and "x \<sharp> P'"   and "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "\<And>M xvec yvec y N P'. \<lbrakk>\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*(xvec @ yvec)\<rparr>\<langle>([(x, y)] \<bullet> N)\<rangle> \<prec> [(x, y)] \<bullet> P'; y \<in> supp N; x \<sharp> N; x \<sharp> P'; x \<noteq> y; y \<sharp> xvec; y \<sharp> yvec; y \<sharp> M; distinct xvec; distinct yvec; xvec \<sharp>* \<Psi>; y \<sharp> \<Psi>; yvec \<sharp>* \<Psi>; xvec \<sharp>* P; y \<sharp> P; yvec \<sharp>* P; xvec \<sharp>* M; y \<sharp> M; yvec \<sharp>* M; xvec \<sharp>* yvec\<rbrakk> \<Longrightarrow> Prop (M\<lparr>\<nu>*(xvec @ y # yvec)\<rparr>\<langle>N\<rangle>) P'"   and "\<And>P'. \<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P' \<Longrightarrow> Prop \<alpha> (\<lparr>\<nu>x\<rparr>P')" shows "Prop \<alpha> P'"
CHOnextConfig_eq: fixes A :: "('a, 'b, 'c, 'd) CHOAlgorithm_scheme"   and r :: "nat"   and cfg :: "'a \<Rightarrow> 'b"   and HO :: "'a \<Rightarrow> 'a set"   and coord :: "'a \<Rightarrow> 'a"   and cfg' :: "'a \<Rightarrow> 'b" shows "CHOnextConfig A r cfg HO coord cfg' = (\<forall>p. CnextState A r p (cfg p) (HOrcvdMsgs A r p (HO p) cfg) (coord p) (cfg' p))"