hcju/isabelleps huggingface dataset:
simps: fixes a1 :: "'vertex"   and a2 :: "(('vertex \<times> ('form, 'var) out_port) \<times> 'vertex \<times> ('form, 'var) antecedent) stream" shows "forbidden_path a1 a2 = (\<exists>v\<^sub>1 p\<^sub>1 v\<^sub>2 p\<^sub>2 pth. a1 = v\<^sub>2 \<and> a2 = ((v\<^sub>1, p\<^sub>1), v\<^sub>2, p\<^sub>2) ## pth \<and> ((v\<^sub>1, p\<^sub>1), v\<^sub>2, p\<^sub>2) \<in> edges \<and> hyps (nodeOf v\<^sub>1) p\<^sub>1 = None \<and> forbidden_path v\<^sub>1 pth)"
widen_rep_def: fixes p1 :: "int extended \<times> int extended"   and p2 :: "int extended \<times> int extended" shows "widen_rep p1 p2 = (if is_empty_rep p1 then p2 else if is_empty_rep p2 then p1 else let (l1, h1) = p1; (l2, h2) = p2 in (if l2 < l1 then -\<infinity> else l1, if h1 < h2 then \<infinity> else h1))"
assms_5: shows "almost_everywhere (MC.T cfg) (alw (\<lambda>\<omega>. HLD (f -` {u}) (smap state \<omega>) \<longrightarrow> ev (HLD Y) \<omega>))"
append_is_Nil_conv: fixes xs :: "'g list"   and ys :: "'g list" shows "(xs @ ys = []) = (xs = [] \<and> ys = [])"
inj_on_RDirProd_carrier: fixes R :: "('a, 'b) ring_scheme"   and Rs :: "('a, 'c) ring_scheme list" shows "inj_on (\<lambda>(a, as). a # as) (carrier (RDirProd R (RDirProd_list Rs)))"
fplus_maybe_simps_2: fixes ys :: "'c\<cdot>maybe" shows "fplus\<cdot>Nothing\<cdot>ys = ys"
Trans_msg: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> c'"   and "isTrans ev" shows "msgs c i = msgs c' i"
detTAG_def: fixes x :: "'b" shows "detTAG x \<equiv> x"
Invariants_vrestr: fixes A :: "(nat \<Rightarrow> 'a) set" assumes "A \<in> sets Invariants" shows "vimage_restr T A = A"
ir_valid_If: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and c1 :: "com"   and c2 :: "com"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "ir_valid P (IF b THEN c1 ELSE c2) c' Q" shows "ir_valid (\<lambda>s s'. P s s' \<and> bval b s) c1 c' (\<lambda>t t'. Q t t' \<and> (\<exists>s s'. P s s' \<and> (c1, s) \<Rightarrow> t \<and> (c', s') \<Rightarrow> t' \<and> bval b s)) \<and> ir_valid (\<lambda>s s'. P s s' \<and> \<not> bval b s) c2 c' (\<lambda>t t'. Q t t' \<and> (\<exists>s s'. P s s' \<and> (c2, s) \<Rightarrow> t \<and> (c', s') \<Rightarrow> t' \<and> \<not> bval b s))"
last_append: fixes xs :: "'a list"   and ys :: "'a list" shows "List.last (xs @ ys) = (if ys = [] then List.last xs else List.last ys)"
add_left_commute: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> V"   and "y \<in> V"   and "z \<in> V" shows "x + (y + z) = y + (x + z)"
inj_onTr2: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and x :: "'d"   and y :: "'d" assumes "inj_on f A"   and "x \<in> A"   and "y \<in> A"   and "f x \<noteq> f y" shows "x \<noteq> y"
leg_simps_1: shows "C.arr HHH\<mu>\<nu>\<pi>\<rho>.dom.leg0"
assertion_prop: fixes x :: "'a" assumes "x \<in> assertion" shows "x * \<top> \<sqinter> (1::'a) = x"
lv_rel_inner: shows "(inner_lv_rel, (\<bullet>)) \<in> lv_rel \<rightarrow> lv_rel \<rightarrow> rnv_rel"
BSIA_def: fixes \<rho> :: "'b V_rec \<Rightarrow> 'b set"   and \<V> :: "'b V_rec"   and Tr :: "'b list set" shows "BSIA \<rho> \<V> Tr \<equiv> \<forall>\<alpha> \<beta>. \<forall>c\<in>C\<^bsub>\<V>\<^esub>. \<beta> @ \<alpha> \<in> Tr \<and> \<alpha> \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [] \<and> Adm \<V> \<rho> Tr \<beta> c \<longrightarrow> (\<exists>\<alpha>'. \<beta> @ [c] @ \<alpha>' \<in> Tr \<and> \<alpha>' \<upharpoonleft> V\<^bsub>\<V>\<^esub> = \<alpha> \<upharpoonleft> V\<^bsub>\<V>\<^esub> \<and> \<alpha>' \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [])"
Detects_def: fixes A :: "'b set"   and B :: "'b set" shows "A Detects B = Always (- A \<union> B) \<inter> (B \<longmapsto>w A)"
homeomorphic_imp_proper_map: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "homeomorphic_map X Y f" shows "proper_map X Y f"
assms: shows "x \<in>\<^sub>\<circ> vconst_on A c"
finite_set_rel_transfer: fixes s :: "'a set"   and s' :: "'c set"   and R :: "('a \<times> 'c) set" assumes "(s, s') \<in> \<langle>R\<rangle>set_rel"   and "single_valued R"   and "finite s" shows "finite s'"
Lp_subalgebra_3: fixes M :: "'b measure"   and F :: "'b measure"   and p :: "ennreal" assumes "subalgebra M F" shows "space\<^sub>N (\<LL> p (restr_to_subalg M F)) \<subseteq> space\<^sub>N (\<LL> p M)"
image_subset_iff_funcset: fixes F :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and B :: "'a set" shows "(F ` A \<subseteq> B) = (F \<in> A \<rightarrow> B)"
dim_subset: fixes S :: "'b set"   and T :: "'b set" assumes "S \<subseteq> T" shows "vs1.dim S \<le> vs1.dim T"
un_Yield2_def: fixes step :: "('f, 'e) step" shows "un_Yield2 step = (case step of Yield x31 x32 \<Rightarrow> x32)"
realistic_executions_def: fixes kstep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and output_f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and s0 :: "'a"   and current :: "'a \<Rightarrow> 'd"   and cswitch :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and kinvolved :: "'b \<Rightarrow> 'd set"   and ifp :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and vpeq :: "'d \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and AS_set :: "'b list set"   and invariant :: "'a \<Rightarrow> bool"   and AS_precondition :: "'a \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool"   and aborting :: "'a \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool"   and waiting :: "'a \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool"   and set_error_code :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and execs :: "'d \<Rightarrow> 'b list list" assumes "Controllable_Interruptible_Separation_Kernel kstep output_f s0 current cswitch kinvolved ifp vpeq AS_set invariant AS_precondition aborting waiting set_error_code" shows "Controllable_Interruptible_Separation_Kernel.realistic_executions AS_set execs \<equiv> \<forall>d. Controllable_Interruptible_Separation_Kernel.realistic_execution AS_set (execs d)"
inducts: fixes x :: "nat set"   and P :: "nat set \<Rightarrow> bool" assumes "x \<in> i_set_cont_min"   and "\<And>n. P [n\<dots>]"   and "\<And>n d. P [n\<dots>,d]" shows "P x"
ans1: fixes x :: "'a" shows "ad x \<cdot> x = (0::'a)"
extf_ext_irrefl_before_trans: fixes f :: "'s" shows "ext_irrefl_before_trans (extf f)"
ab_eq: fixes a :: "'a"   and b :: "'a" assumes "a \<in> H"   and "b \<in> H" shows "a \<otimes>\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> b = a \<otimes> b"
append_Cons_eq_iff: fixes x :: "'c"   and xs :: "'c list"   and ys :: "'c list"   and xs' :: "'c list"   and ys' :: "'c list" assumes "x \<notin> set xs"   and "x \<notin> set ys" shows "(xs @ x # ys = xs' @ x # ys') = (xs = xs' \<and> ys = ys')"
col: shows "swapcols k l A = A * swaprows_mat n k l"
run_owner_v_def: fixes n :: "nat"   and s :: "(agent \<times> msg) set" shows "run_owner_v n s \<equiv> stage_owner_v n (run_asset_iv n s)"
mult_less_cancel_left_pos: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and abs :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d"   and a :: "'d"   and b :: "'d" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times"   and "less zero c" shows "less (times c a) (times c b) = less a b"
steps: shows "(ts, m, \<S>) \<Rightarrow>\<^sub>d\<^sup>* (ts'', m'', \<S>'')"
list_orderE: fixes ops :: "('a \<times> 'a option) list"   and x :: "'a"   and y :: "'a" assumes "list_order ops x y" shows "\<exists>xs ys zs. interp_ins ops = xs @ [x] @ ys @ [y] @ zs"
set_mkIde: fixes S :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A :: "'a set" assumes "replete_set_category S"   and "A \<subseteq> Collect (category.terminal S)" shows "set_category_data.set S (set_category.some_img S (cardSuc (cmax |UNIV| natLeq))) (set_category_given_img.mkIde S (set_category.some_img S (cardSuc (cmax |UNIV| natLeq))) (cardSuc (cmax |UNIV| natLeq)) A) = A"
dur_pos: fixes xs :: "('c \<Rightarrow> real) stream"   and i :: "nat" shows "0 \<le> dur xs i"
2: shows "infs (degen cs) (stl w ||| count cs (shd w) k ## sscan (count cs) (stl w) (count cs (shd w) k))"
comp_char: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" shows "CC.comp g f = (if seq (fst g) (fst f) \<and> seq (snd g) (snd f) then (fst g \<cdot> fst f, snd g \<cdot> snd f) else (null, null))"
M5exist_var3: fixes k :: "'b"   and l :: "'b"   and q :: "'b"   and t :: "'b"   and r :: "'b" assumes "k \<parallel> l"   and "l \<parallel> q"   and "q \<parallel> t"   and "t \<parallel> r" shows "\<exists>lqt. k \<parallel> lqt \<and> lqt \<parallel> r"
atom_map_sup_pres: fixes x :: "'a"   and y :: "'a" shows "atom_map (x \<squnion> y) = atom_map x \<union> atom_map y"
not_subtypes_aux: fixes CT :: "nat \<Rightarrow> classDef option"   and C :: "nat"   and Da :: "nat"   and CDef :: "classDef"   and D :: "nat" assumes "CT \<turnstile> C <: Da"   and "C \<noteq> Da"   and "CT C = Some CDef"   and "cSuper CDef = D" shows "CT \<turnstile> D <: Da"
imp_false: fixes H :: "'b pl set"   and p :: "'b pl"   and q :: "'b pl" assumes "H |- p"   and "H |- q -> false" shows "H |- (p -> q) -> false"
from_list_refine: fixes \<alpha>w :: "'a \<Rightarrow> 'b set \<Rightarrow> nat"   and \<alpha>g :: "'a \<Rightarrow> 'b ugraph"   and invar :: "'a \<Rightarrow> bool"   and adj :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> nat) list"   and empty :: "'a"   and add_edge :: "'b \<times> 'b \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and l :: "(('b \<times> 'b) \<times> nat) list" assumes "adt_wgraph \<alpha>w \<alpha>g invar adj empty add_edge"   and "valid_wgraph_repr l" shows "invar (adt_wgraph.from_list empty add_edge l) \<and> \<alpha>g (adt_wgraph.from_list empty add_edge l) = graph_from_list (map fst l) \<and> \<alpha>w (adt_wgraph.from_list empty add_edge l) = weight_from_list l"
prefix_code_1: fixes xs :: "'c list" shows "prefix [] xs = True"
continuous_on_interval_bij: fixes s :: "'a set"   and a :: "'a"   and b :: "'a"   and u :: "'a"   and v :: "'a" shows "continuous_on s (interval_bij (a, b) (u, v))"
chain_1: shows "\<forall>x y. R x y \<longrightarrow> R' (f x) (f y)"
\<alpha>_MST_def: fixes \<pi>i :: "'m" shows "\<alpha>_MST \<pi>i \<equiv> graph {r} {(u, v) |u v. M_lookup \<pi>i u = Some v}"
closed_sequential_limits: fixes S :: "'d set" shows "closed S = (\<forall>x l. (\<forall>n. x n \<in> S) \<and> x \<longlonglongrightarrow> l \<longrightarrow> l \<in> S)"
mem_carrier: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "submonoid H G"   and "x \<in> H" shows "x \<in> carrier G"
BCh_def: shows "BCh \<equiv> map_of_default (\<lambda>_. {}) [(Xh d1h1, \<lambda>X. set_option (find (\<lambda>x. x \<in> X) [d1h1, d2h1, d3h1])), (Xh d1h2, \<lambda>X. set_option (find (\<lambda>x. x \<in> X) [])), (Xh d1h3, \<lambda>X. set_option (find (\<lambda>x. x \<in> X) [d3h3, d2h3, d1h3]))]"
QR_nonempty: shows "Q \<inter> R \<noteq> {}"
null_eqI: fixes n :: "'a" assumes "\<And>f. Right.comp n f = n \<and> Right.comp f n = n" shows "n = Right.null"
span_is_subset2: fixes R :: "('a, 'b) ring_scheme"   and M :: "('a, 'c, 'd) module_scheme"   and S :: "'c set" assumes "Module.module R M"   and "S \<subseteq> carrier M" shows "LinearCombinations.module.span R M S \<subseteq> carrier M"
Prim_Invar1_loc_def: fixes w :: "'a set \<Rightarrow> nat"   and g :: "'a ugraph"   and r :: "'a"   and A :: "('a \<times> 'a) set" shows "Prim_Invar1_loc w g r A \<equiv> Prim.prim_invar1 w g r A"
c4: assumes "ip \<in> kD rt"   and "sqn rt ip = \<pi>\<^sub>2 r"   and "the (flag rt ip) = Aodv_Basic.inv" shows "P (rt(ip \<mapsto> addpre r (\<pi>\<^sub>7 (the (rt ip)))))"
dom_simp: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Arr :: "'b \<Rightarrow> bool"   and f :: "'b" assumes "subcategory C Arr"   and "partial_magma.arr (subcategory.comp C Arr) f" shows "partial_magma.dom (subcategory.comp C Arr) f = partial_magma.dom C f"
second_vertices_def: shows "second_vertices \<equiv> second_vertex ` paths"
ind_hyp: shows "insert_sort_insert_by cmp e l' <~~> e # l'"
sing: fixes x :: "'a" assumes "x \<in> S" shows "P (frag_of x)"
iso_is_retraction: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C"   and "category.iso C f" shows "category.retraction C f"
simps_4: fixes xs :: "'b llist" assumes "\<not> lnull xs" shows "ltl (lalternate xs) = (case xs of LCons x xs \<Rightarrow> lalternate (ltl xs))"
countable_mult: fixes A :: "V"   and B :: "V" assumes "countable (elts A)"   and "countable (elts B)" shows "countable (elts (A * B))"
Guard: fixes s :: "'d"   and g :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c :: "('d, 'e, 'f) com"   and n :: "nat"   and t :: "('d, 'f) xstate"   and f :: "'f" assumes "s \<in> g"   and "\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> =n\<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>Guard f g c,Normal s\<rangle> =n\<Rightarrow> t"
udvdE: fixes v :: "'b word"   and w :: "'b word" assumes "v udvd w" obtains u :: "'b word" where "unat w = unat v * unat u"
mult_zero_left: fixes z :: "complex_homo" assumes "z \<noteq> \<infinity>\<^sub>h" shows "0\<^sub>h *\<^sub>h z = 0\<^sub>h"
Diff_insert_absorb: fixes x :: "'d"   and A :: "'d set" assumes "x \<notin> A" shows "insert x A - {x} = A"
split_max_height: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and k :: "nat"   and t :: "'b btree"   and sub :: "'b btree"   and sep :: "'b" assumes "split split"   and "split.split_max k t = (sub, sep)"   and "nonempty_lasttreebal t"   and "t \<noteq> Leaf" shows "height sub = height t"
simps: fixes a1 :: "('c \<times> 'd) list"   and a2 :: "'d \<Rightarrow> 'c set" shows "crdt_ops a1 a2 = ((\<exists>deps. a1 = [] \<and> a2 = deps) \<or> (\<exists>xs deps oid oper. a1 = xs @ [(oid, oper)] \<and> a2 = deps \<and> crdt_ops xs deps \<and> oid \<notin> set (map fst xs) \<and> (\<forall>ref\<in>deps oper. ref \<in> set (map fst xs) \<and> ref < oid)))"
vcomp_vimage_vtimes_right: fixes r :: "V"   and Y :: "V"   and Z :: "V"   and X :: "V" assumes "r `\<^sub>\<circ> Y = Z" shows "r \<circ>\<^sub>\<circ> X \<times>\<^sub>\<circ> Y = X \<times>\<^sub>\<circ> Z"
Ex_rev_wrap_def: fixes P :: "'b list \<Rightarrow> bool" shows "Ex_rev_wrap P = (\<exists>x. P (rev x))"
R: shows "(\<lambda>(b, k). (Inum (a # bs) b, k)) ` R = (\<lambda>(b, k). (Inum (a # bs) b, k)) ` set (\<rho> p)"
assms_2: shows "p\<^sub>2 = Point x\<^sub>2 y\<^sub>2"
independent_card_le_aff_dim: fixes B :: "'a set"   and V :: "'a set" assumes "B \<subseteq> V"   and "\<not> affine_dependent B" shows "int (card B) \<le> aff_dim V + 1"
simps_6: fixes a1 :: "'a gexp"   and a2 :: "'a gexp"   and s :: "'a \<Rightarrow> value option" shows "gval (Nor a1 a2) s = \<not>? gval a1 s \<or>? gval a2 s"
f_Exec_Comp_N_def: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and n :: "nat"   and xs :: "'e list"   and c :: "'d" shows "f_Exec_Comp_N trans_fun n xs c \<equiv> f_Exec_Comp trans_fun (xs \<down> n) c"
as\<^sub>0_def: shows "as\<^sub>0 \<equiv> (0::'a, 0::'b, \<lambda>_. INIT)"
rel: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" assumes "wls s X"   and "(X, Y) \<in> rel s" shows "wls s Y \<and> skel Y = skel X"
dBox_def: fixes A :: "'b hrel"   and \<Phi> :: "'b upred" shows "\<^bold>[A\<^bold>]\<Phi> = A wp \<Phi>"
square_free_multD_3: fixes f :: "'a poly"   and g :: "'a poly" assumes "square_free (f * g)" shows "square_free g"
correctCompositionDiffLevelsS6: shows "correctCompositionDiffLevels sS6"
pred_mono: fixes P :: "'a \<Rightarrow> bool"   and Pa :: "'a \<Rightarrow> bool" assumes "P \<le> Pa" shows "list_all P \<le> list_all Pa"
function_ring_def: fixes S :: "'d set"   and R :: "('e, 'f) ring_scheme" shows "function_ring S R = \<lparr>carrier = S \<rightarrow>\<^sub>E carrier R, mult = function_mult S R, one = function_one S R, zero = function_zero S R, add = function_add S R, smult = function_scalar_mult S R\<rparr>"
intros_10: fixes F :: "'b formula" shows "(\<^bold>\<not> F \<^bold>\<rightarrow> \<bottom>) \<^bold>\<rightarrow> F \<in> AX10"
pmf_Pi_empty: fixes dflt :: "'b"   and p :: "'a \<Rightarrow> 'b pmf" shows "Pi_pmf {} dflt p = return_pmf (\<lambda>_. dflt)"
prob_compl: fixes A :: "'a set" assumes "A \<in> events" shows "prob (space M - A) = 1 - prob A"
run_one_step_basic_select_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, $Select) = (s', vs', res)" shows "(\<exists>r. res = RSNormal r) \<or> (\<exists>e. res = RSCrash e)"
distinct_fst_polychain_of: fixes xs :: "(real \<times> real) list"   and x0 :: "real \<times> real" assumes "list_all (\<lambda>x. x \<noteq> 0) xs"   and "list_all (\<lambda>x. Counterclockwise_2D_Arbitrary.lex x 0) xs" shows "distinct (map fst (polychain_of x0 xs))"
mono_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'c" assumes "class.order less_eq less" shows "order.mono less_eq f = (\<forall>x y. less_eq x y \<longrightarrow> f x \<le> f y)"
dioph_empt_def: fixes ks :: "int list"   and l :: "int" shows "dioph_empt ks l = (replicate (nat (2 * max \<bar>l\<bar> (sum_list (map abs ks)) + 1)) None, [])"
S5n_subjective_eq: fixes M :: "('b, 'c, 'd) KripkeStructure"   and a :: "'b"   and \<phi> :: "('b, 'c) Kform"   and w :: "'d"   and w' :: "'d" assumes "S5n M"   and "subjective a \<phi>"   and "(w, w') \<in> relations M a" shows "M, w \<Turnstile> \<phi> = M, w' \<Turnstile> \<phi>"
InitBlockReds_aux: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   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 V :: "char list"   and T :: "ty" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>" shows "\<forall>h l sh h' l' sh' v. s = (h, l(V \<mapsto> v), sh) \<longrightarrow> s' = (h', l', sh') \<longrightarrow> P \<turnstile> \<langle>{V:T; V:=Val v;; e},(h, l, sh),b\<rangle> \<rightarrow>* \<langle>{V:T; V:=Val (the (l' V));; e'},(h', l'(V := l V), sh'),b'\<rangle>"
sset_sproduct: fixes s1 :: "'a stream"   and s2 :: "'b stream" shows "i.R (sproduct s1 s2) = i.R s1 \<times> i.R s2"
piecewise_continuous_on_combine: fixes a :: "'c"   and b :: "'c"   and J :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and c :: "'c" assumes "piecewise_continuous_on a b J f"   and "piecewise_continuous_on b c J f" shows "piecewise_continuous_on a c J f"
lm095: fixes R :: "('a \<times> 'b) set" shows "R \<subseteq> Domain R \<times> Range R"
is_sublist_rec: fixes xs :: "'a list"   and ys :: "'a list" shows "is_sublist xs ys = (if |ys| < |xs| then False else if xs = take |xs| ys then True else is_sublist xs (tl ys))"
cqt_basic_4: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>"   and \<psi> :: "'a \<Rightarrow> \<o>" shows "[(\<^bold>\<forall>\<alpha>. \<phi> \<alpha> \<^bold>& \<psi> \<alpha>) \<^bold>\<equiv> (\<^bold>\<forall>\<alpha>. \<phi> \<alpha>) \<^bold>& (\<^bold>\<forall>\<alpha>. \<psi> \<alpha>) in v]"
simps_3: fixes p :: "nat"   and a :: "floatarith"   and vs :: "((real \<times> real pdevs) \<times> real) list" shows "approx_floatarith p (floatarith.Inverse a) vs = approx_floatarith p a vs \<bind> inverse_aform_err p"
create_gba_acc: fixes \<phi> :: "'b ltlr" shows "create_gba \<phi> \<le> SPEC (\<lambda>\<A>. \<forall>\<xi>. gba.accept \<A> \<xi> = \<xi> \<Turnstile>\<^sub>r \<phi>)"
hseq_char': fixes \<nu> :: "'b"   and \<mu> :: "'b" shows "hseq \<nu> \<mu> = (\<nu> \<star> \<mu> \<noteq> null)"
simps_2: fixes a :: "'c set"   and b :: "'c set" shows "(RES a \<le> RES b) = (a \<subseteq> b)"
TC_Sup_distrib: fixes X :: "V set" assumes "small X" shows "TC (\<Squnion> X) = \<Squnion> (TC ` X)"
as: fixes x :: "real"   and xa :: "real" assumes "x \<in> {- 1..1}"   and "xa \<in> {- 1..1}" shows "f x \<noteq> g xa"
pdevs_eqI: fixes x :: "'b pdevs"   and y :: "'b pdevs" assumes "\<And>i. pdevs_apply x i = pdevs_apply y i" shows "x = y"
plossless_gpv_mk_lossless_gpv: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and \<I>' :: "('a, 'b) \<I>"   and x :: "'c" assumes "plossless_gpv \<I> gpv"   and "\<I> \<turnstile>g gpv \<surd>"   and "outs_\<I> \<I> = outs_\<I> \<I>'" shows "plossless_gpv \<I>' (mk_lossless_gpv (responses_\<I> \<I>) x gpv)"
back_edge_disc_lt_fin: fixes v :: "'v"   and w :: "'v" assumes "(v, w) \<in> back_edges s"   and "v \<in> dom (finished s)" shows "\<delta> s w < \<phi> s v"
CanonicalComp_def: fixes T :: "(ZF, ZF) Axioms"   and f :: "ZF"   and g :: "ZF" shows "CanonicalComp T f g \<equiv> THE h. \<exists>e e'. h = [CLDomain T f,sub e in e',CLCodomain T g]\<^bsub>T\<^esub> \<and> f = [CLDomain T f,e,CLCodomain T f]\<^bsub>T\<^esub> \<and> g = [CLDomain T g,e',CLCodomain T g]\<^bsub>T\<^esub>"
AnnAwait: fixes c :: "'a com"   and r :: "'a set"   and b :: "'a set"   and q :: "'a set" assumes "atom_com c"   and "\<parallel>- (r \<inter> b) c q" shows "\<turnstile> AnnAwait r b c  q"
diagonal_to_Smith_aux_append: fixes A :: "(('a, 'b) vec, 'c) vec"   and xs :: "nat list"   and ys :: "nat list"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "diagonal_to_Smith_aux A (xs @ ys) bezout = diagonal_to_Smith_aux (diagonal_to_Smith_aux A xs bezout) ys bezout"
prems_1: shows "g \<turnstile> ns @ [n]\<Down>s"
M'_closes: shows "decomp_closure M {t. M \<turnstile> t \<and> (\<exists>t'\<in>M. t \<sqsubseteq> t')}"
mat_diag_one: fixes n :: "nat" shows "mat_diag n (\<lambda>x. 1::'a) = 1\<^sub>m n"
through: shows "blinfun_apply (Ds (poincare_map {x \<in> S. s x = 0} x)) (f (poincare_map {x \<in> S. s x = 0} x)) \<noteq> 0"
vsubst_trans: fixes X :: "('c, 'd, 'varSort, 'var, 'e) qTerm set"   and ys :: "'varSort"   and y1 :: "'var"   and y2 :: "'var"   and y :: "'var" assumes "good X"   and "fresh ys y1 X" shows "X #[y1 // y]_ys #[y2 // y1]_ys = X #[y2 // y]_ys"
simps_1: fixes q :: "hf" shows "Power.nextl q [] = q"
valid_W_inv_no_mark_stores_invD: fixes s :: "'a process_name \<Rightarrow> ('b, 'a, 'c, 'd) Model.local_state"   and p :: "'a process_name" assumes "sys_mem_lock s \<noteq> Some p"   and "valid_W_inv s" shows "filter is_mw_Mark (sys_mem_store_buffers p s) = []"
oalist_inv_rawI: fixes xs :: "('a \<times> 'c) list" assumes "(0::'c) \<notin> snd ` set xs"   and "sorted_wrt lt (map fst xs)" shows "oalist_inv_raw xs"
measurable_insertion: fixes p :: "real mpoly"   and n :: "nat" assumes "vars p \<subseteq> {..<n}" shows "(\<lambda>f. insertion f p) \<in> borel_measurable (lborel_f n)"
bpE_4: fixes P :: "'a set set" assumes "bp P" shows "\<forall>B\<in>P. W B \<le> real c"
View'_def: shows "view_merkle.View' = (\<lambda>((vm, vd), vs). View vm vd vs)"
VectorSpaceHom: shows "VectorSpaceHom fsmult' V fsmult' T"
hyps_2: shows "liftAll (\<lambda>a. qFreshAbs ys (y @ys[z1 \<and> z2]_zs) (a $[[z1 \<and> z2]]_zs) = qFreshAbs ys y a) binp"
edges_of_comp1: fixes xs :: "'a list"   and ys :: "'a list" shows "H.edges_of_walk xs \<subseteq> H.edges_of_walk (xs @ ys)"
RModHom_composite_left: fixes N :: "'w set"   and smult'' :: "('f, 'g) aezfun \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'w \<Rightarrow> 'b" assumes "ImG \<subseteq> N"   and "RModuleHom FG (\<star>) N smult'' S" shows "RModuleHom FG (\<cdot>) V smult'' (S \<circ> T)"
I: shows "((\<lambda>x. x powr (z + 1) / (exp x - 1) - x powr z / exp x) has_integral Gamma (z + 2) * (Re (zeta (complex_of_real (z + 2))) - 1 / (z + 1))) {0<..}"
cdf_lim_infty: shows "(\<lambda>i. cdf M (real i)) \<longlonglongrightarrow> prob (space M)"