hcju/isabelleps huggingface dataset:
divide_le_0_1_iff: fixes a :: "'a" shows "((1::'a) / a \<le> (0::'a)) = (a \<le> (0::'a))"
sum_nonneg_eq_0_iff: 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 A :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq zero (f x)" shows "(comm_monoid_add.sum plus zero f A = zero) = (\<forall>x\<in>A. f x = zero)"
min_inf: fixes f :: "atom fm"   and xs :: "real list" assumes "nqfree f" shows "\<exists>x. \<forall>y\<le>x. R.I (inf\<^sub>- f) xs = R.I f (y # xs)"
assms: shows "x \<in> set (factorization_lattice u d m)"
opt_bst_wpl2_dom: shows "All opt_bst_wpl2_dom"
comp_proj_assoc_1: fixes a0 :: "'a"   and a1 :: "'a"   and a2 :: "'a" assumes "ide a0"   and "ide a1"   and "ide a2" shows "pr1 a0 (local.prod a1 a2) \<cdot> assoc a0 a1 a2 = pr1 a0 a1 \<cdot> pr1 (local.prod a0 a1) a2"
closure_of_interior_of: fixes X :: "'c topology"   and S :: "'c set" shows "X closure_of S = topspace X - X interior_of (topspace X - S)"
finite_gpv_induct: fixes \<I> :: "('a, 'b) \<I>"   and x :: "('c, 'a, 'b) gpv"   and P :: "('c, 'a, 'b) gpv \<Rightarrow> bool" assumes "finite_gpv \<I> x"   and "\<And>gpv. \<lbrakk>\<And>out c input. \<lbrakk>IO out c \<in> set_spmf (the_gpv gpv); input \<in> responses_\<I> \<I> out\<rbrakk> \<Longrightarrow> finite_gpv \<I> (c input); \<And>out c input. \<lbrakk>IO out c \<in> set_spmf (the_gpv gpv); input \<in> responses_\<I> \<I> out\<rbrakk> \<Longrightarrow> P (c input)\<rbrakk> \<Longrightarrow> P gpv" shows "P x"
lset_lconcat_lfinite: fixes xss :: "'a llist llist" assumes "\<forall>xs\<in>lset xss. lfinite xs" shows "lset (lconcat xss) = \<Union> (lset ` lset xss)"
k3: fixes i :: "nat" assumes "k \<le> i \<and> i < n" shows "\<not> P ((A @ B) ! i)"
simps: fixes ant :: "form multiset"   and suc :: "form multiset" shows "antec ( ant \<Rightarrow>* suc) = ant"
sarp_on_def: fixes A :: "'b set"   and f :: "'b set \<Rightarrow> 'b set" shows "sarp_on A f = (\<forall>(x, y)\<in>(rsp_on A f)\<^sup>+. (y, x) \<notin> rwp_on A f)"
times_1_both: fixes z :: "int"   and z' :: "int" assumes "0 < z"   and "z * z' = 1" shows "z = 1 \<and> z' = 1"
RSubmodule_RSpan_single: fixes m :: "'m" assumes "m \<in> M" shows "RSubmodule (RSpan [m])"
r_min_eq_r_min_opt: fixes t :: "('b, unit) RBT.rbt" assumes "\<not> RBT.is_empty t" shows "r_min t = r_min_opt t"
code_unique_dec: fixes us :: "'a list list" assumes "us \<in> lists \<C>" shows "Dec \<C> concat us = us"
reflclp_tranclp: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "r\<^sup>+\<^sup>+\<^sup>=\<^sup>= = r\<^sup>*\<^sup>*"
distinct_35: fixes x41 :: "('c, 'd) Term.term"   and x42 :: "('c, 'd) Term.term"   and x71 :: "'d list"   and x72 :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list"   and x73 :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list" shows "insert\<langle>x41,x42\<rangle> \<noteq> \<forall>x71\<langle>\<or>\<noteq>: x72 \<or>\<notin>: x73\<rangle>"
Elem_def: fixes uu :: "String.literal"   and uv :: "(String.literal \<times> String.literal) list"   and uw :: "xml_tree list" shows "xml.Elem uu uv uw = XML_Tree"
le_num_One_iff: fixes x :: "num" shows "(x \<le> num.One) = (x = num.One)"
independent_explicit: fixes B :: "'a set" shows "independent B = (finite B \<and> (\<forall>c. (\<Sum>v\<in>B. c v *\<^sub>R v) = (0::'a) \<longrightarrow> (\<forall>v\<in>B. c v = 0)))"
arr_MkArr: fixes A :: "'obj"   and B :: "'obj"   and f :: "'arr list" assumes "A \<in> Obj"   and "B \<in> Obj"   and "f \<in> Collect (G.path A B)" shows "arr (MkArr A B f)"
subset_emptyI: fixes A :: "'a set" assumes "\<And>x. x \<in> A \<Longrightarrow> False" shows "A \<subseteq> {}"
put_def: fixes s :: "'c" shows "put s \<equiv> \<lambda>_. ({((), s)}, False)"
power_Suc: fixes one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and n :: "nat" shows "power.power one times a (Suc n) = times a (power.power one times a n)"
someI2_ex: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" assumes "\<exists>a. P a"   and "\<And>x. P x \<Longrightarrow> Q x" shows "Q (SOME x. P x)"
elims: fixes x :: "(int \<times> int) list"   and y :: "(int \<times> int) \<times> int \<times> int" assumes "closest_pair x = y" obtains "x = []" and "y = undefined"   | uu_ :: "int \<times> int" where "x = [uu_]" and "y = undefined"   | v :: "int \<times> int" and vb :: "int \<times> int" and vc :: "(int \<times> int) list" where "x = v # vb # vc" and "y = (let (uu_, p) = closest_pair_rec (mergesort fst (v # vb # vc)) in p)"
strict_trans2: fixes a :: "'x \<Rightarrow>\<^sub>0 nat"   and b :: "'x \<Rightarrow>\<^sub>0 nat"   and c :: "'x \<Rightarrow>\<^sub>0 nat" assumes "a \<prec> b"   and "b \<preceq> c" shows "a \<prec> c"
sgn_div: fixes y :: "'a"   and x :: "'a" assumes "y \<noteq> (0::'a)"   and "sgn x = sgn y" shows "(0::'a) < x / y"
def_scheme: fixes x :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>" shows "f x \<equiv> \<lambda>\<tau>. if (\<delta> x) \<tau> = true \<tau> then g (x \<tau>) else invalid \<tau>"
transpose_mult_matrix: fixes fmul :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and fadd :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'b matrix"   and B :: "'b matrix" assumes "\<forall>a. fmul (0::'b) a = (0::'c)"   and "\<forall>a. fmul a (0::'b) = (0::'c)"   and "fadd (0::'c) (0::'c) = (0::'c)"   and "\<forall>x y. fmul y x = fmul x y" shows "transpose_matrix (mult_matrix fmul fadd A B) = mult_matrix fmul fadd (transpose_matrix B) (transpose_matrix A)"
count_diff: fixes M :: "'d multiset"   and N :: "'d multiset"   and a :: "'d" shows "count (M - N) a = count M a - count N a"
rev_append_def: shows "rev_append \<equiv> rec_list (\<lambda>ac. ac) (\<lambda>x xs xsa ac. xsa (x # ac))"
prob_space_distr: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> 'e"   and M' :: "'e measure" assumes "prob_space M"   and "f \<in> M \<rightarrow>\<^sub>M M'" shows "prob_space (distr M M' f)"
Conj_eqvt: fixes xset :: "('e, 'f, 'g) formula set['e]"   and p :: "perm" assumes "finite (supp xset)" shows "p \<bullet> Conj xset = Conj (p \<bullet> xset)"
arr_iff_has_src: fixes \<mu> :: "'a" shows "arr \<mu> = (sources \<mu> \<noteq> {})"
index_conv_size_if_notin: fixes x :: "'b"   and xs :: "'b list" assumes "x \<notin> set xs" shows "index xs x = length xs"
scopep_scope_eq: fixes outPorts :: "'b \<Rightarrow> 'c fset"   and vertices :: "'a fset"   and nodeOf :: "'a \<Rightarrow> 'b"   and edges :: "(('a \<times> 'c) \<times> 'a \<times> 'd) set"   and ps :: "'a \<times> 'd" shows "Pre_Port_Graph.scopep outPorts vertices nodeOf edges ps = (\<lambda>x. x \<in> Pre_Port_Graph.scope outPorts vertices nodeOf edges ps)"
multiset_filter_subset: fixes f :: "'b \<Rightarrow> bool"   and M :: "'b multiset" shows "filter_mset f M \<subseteq># M"
cos_add: fixes x :: "'b"   and y :: "'b" shows "cos (x + y) = cos x * cos y - sin x * sin y"
InterI: fixes C :: "'c set set"   and A :: "'c" assumes "\<And>X. X \<in> C \<Longrightarrow> A \<in> X" shows "A \<in> \<Inter> C"
assms_3: shows "\<bar>int (size l) - int (size r)\<bar> \<le> 1"
intro: fixes is_open :: "'a set \<Rightarrow> bool"   and U :: "'a set" assumes "is_open U" shows "open_cover_of_open_subset_axioms is_open U"
simps_2: fixes x :: "brick list"   and xs :: "wall"   and y :: "brick list" shows "x * xs \<otimes> basic y = (if codomain_block y = 0 then (x \<otimes> y) * xs else (x \<otimes> y) * (xs \<otimes> basic (make_vert_block (nat (codomain_block y)))))"
wt_hd_le: fixes s :: "('s, 'v) tm" shows "wt (Hd (head s)) \<le> wt s"
not_less_iff_gr_or_eq: fixes y :: "'a list"   and x :: "'a list" shows "(\<not> y <lex x) = (x <lex y \<or> x = y)"
finite_measure_count_space: fixes A :: "'b set" assumes "finite A" shows "finite_measure (count_space A)"
comm_semiring_hom_axioms: fixes p :: "'b poly" shows "comm_semiring_hom (\<lambda>q. q \<circ>\<^sub>p p)"
Method_def: fixes P :: "'a prog"   and C :: "String.literal"   and M :: "String.literal"   and Ts :: "ty list"   and T :: "ty"   and m :: "'a option"   and D :: "String.literal" shows "P \<turnstile> C sees M: Ts\<rightarrow>T = m in D \<equiv> \<exists>Mm. P \<turnstile> C sees_methods Mm \<and> Mm M = \<lfloor>((Ts, T, m), D)\<rfloor>"
prod_emb_def: fixes I :: "'c set"   and M :: "'c \<Rightarrow> 'd measure"   and K :: "'c set"   and X :: "('c \<Rightarrow> 'd) set" shows "prod_emb I M K X = (\<lambda>x. restrict x K) -` X \<inter> (\<Pi>\<^sub>E i\<in>I. space (M i))"
transPrecedesOrder: fixes list :: "'b list" shows "trans (precedesOrder list)"
in_components_nonempty: fixes c :: "'a set"   and s :: "'a set" assumes "c \<in> components s" shows "c \<noteq> {}"
bvars\<^sub>s\<^sub>t_def: fixes S :: "('d, 'c) strand_step list" shows "bvars\<^sub>s\<^sub>t S \<equiv> \<Union> (set (map (set \<circ> bvars\<^sub>s\<^sub>t\<^sub>p) S))"
prodag_comp_i: fixes a :: "'c \<Rightarrow> 'd"   and I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and i :: "'c" assumes "a \<in> carr_prodag I A"   and "i \<in> I" shows "a i \<in> carrier (A i)"
unat_arith_simps_2: fixes a :: "'b word"   and b :: "'b word" shows "(a < b) = (unat a < unat b)"
awlast_append: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and q :: "'d list" assumes "wf_digraph G" shows "last (pre_digraph.awalk_verts G u (p @ q)) = last (pre_digraph.awalk_verts G (last (pre_digraph.awalk_verts G u p)) q)"
sym_Un: fixes r :: "('c \<times> 'c) set"   and s :: "('c \<times> 'c) set" assumes "sym r"   and "sym s" shows "sym (r \<union> s)"
reduced_word_forI_length': fixes as :: "'a list"   and A :: "'a set"   and a :: "'a" assumes "as \<in> lists A"   and "sum_list as = a"   and "order.greater_eq (word_length A a) (length as)" shows "reduced_word_for A a as"
inj_on_to_nat_on: fixes A :: "'c set" assumes "countable A" shows "inj_on (to_nat_on A) A"
f_def: shows "f \<equiv> \<lambda>z u. complex_of_real (- ln u) powr z / complex_of_real (1 - u) * complex_of_real (- ln u + u - 1)"
fsplit: shows "infsetsum f A = Complex (\<Sum>\<^sub>ax\<in>A. Re (f x)) (\<Sum>\<^sub>ax\<in>A. Im (f x))"
lookup_restrictpm: fixes A :: "'b set"   and m :: "'b \<Rightarrow>\<^sub>0 'a"   and x :: "'b" shows "lookup (restrictpm A m) x = (if x \<in> A then lookup m x else (0::'a))"
vec_nth_inject: fixes x :: "('e, 'd) vec"   and y :: "('e, 'd) vec" shows "(($) x = ($) y) = (x = y)"
lzip_eq_LCons_conv: fixes xs :: "'c llist"   and ys :: "'d llist"   and z :: "'c \<times> 'd"   and zs :: "('c \<times> 'd) llist" shows "(lzip xs ys = LCons z zs) = (\<exists>x xs' y ys'. xs = LCons x xs' \<and> ys = LCons y ys' \<and> z = (x, y) \<and> zs = lzip xs' ys')"
dom_cod: fixes f :: "'c local.term set" assumes "\<F>C.arr f" shows "\<F>C.dom (\<F>C.cod f) = \<F>C.cod f"
countable_UN: fixes I :: "'b set"   and A :: "'b \<Rightarrow> 'c set" assumes "countable I"   and "\<And>i. i \<in> I \<Longrightarrow> countable (A i)" shows "countable (\<Union> (A ` I))"
ae_f: shows "AE x in f. f.prob {x} \<noteq> 0"
t_twice_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_twice stp = (Suc t_twice_len, Bk \<up> ln @ Bk # Bk # ires, Oc \<up> Suc (2 * rs) @ Bk \<up> rn)"
prodag_pOp: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)" shows "(\<plusminus>\<^bsub>a\<Pi>\<^bsub>I\<^esub> A\<^esub>) = prod_pOp I A"
subst_def: shows "subst \<equiv> \<lambda>x y t'. lam_rec (\<lambda>x. if x = y then t' else Var x) (\<lambda>t1 t2. App) (\<lambda>x t. Lam x) x"
A1a: shows "\<lfloor>(\<lambda>w. \<forall>x. (\<P> (\<rightharpoondown>x) \<^bold>\<rightarrow> \<^bold>\<not>\<P> x) w)\<rfloor>"
map_pmf_ident: shows "map_pmf (\<lambda>x. x) = (\<lambda>x. x)"
igSwapAbsIPresIGWlsAbsSTR_def: fixes MOD :: "('h, 'i, 'j, 'k, 'l, 'm, 'n, 'o, 'p) model_scheme" shows "igSwapAbsIPresIGWlsAbsSTR MOD \<equiv> \<forall>zs z1 z2 us s A. igWlsAbs MOD (us, s) (igSwapAbs MOD zs z1 z2 A) = igWlsAbs MOD (us, s) A"
psiI: shows "\<turnstile> psi \<longrightarrow> \<box>I"
in_atlas_order_le: fixes l :: "enat"   and c :: "('a, 'e) chart" assumes "l \<le> l"   and "c \<in> l.src.atlas" shows "c \<in> c_manifold.atlas charts1 l"
errMOD_igWlsAbsDisj: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h) model" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.igWlsAbsDisj Delta MOD" shows "FixSyn.igWlsAbsDisj Delta (FixSyn.errMOD Delta MOD)"
assms_4: shows "gt (A $$ (i, j)) (B $$ (i, j))"
LIM_zero: fixes f :: "'c \<Rightarrow> 'd"   and l :: "'d"   and F :: "'c filter" assumes "(f \<longlongrightarrow> l) F" shows "((\<lambda>x. f x - l) \<longlongrightarrow> (0::'d)) F"
preserves_def: fixes v :: "'c \<Rightarrow> 'd" shows "preserves v \<equiv> \<Inter>z. stable {s. v s = z}"
min_wpl_simps_1: fixes j :: "int"   and i :: "int" assumes "j < i" shows "local.min_wpl i j = 0"
not_Interp_imp_not_INTERP: fixes S :: "'a literal multiset \<Rightarrow> 'a literal multiset"   and D' :: "'a literal multiset"   and D :: "'a literal multiset"   and N :: "'a literal multiset set" assumes "ground_resolution_with_selection S"   and "D' \<le> D"   and "D' \<in> N \<and> S D' = {#} \<or> Max (atms_of D') < Max (atms_of D)"   and "\<not> ground_resolution_with_selection.Interp S N D \<Turnstile> D'" shows "\<not> ground_resolution_with_selection.INTERP S N \<Turnstile> D'"
find_first_in_map: fixes xs :: "'a list"   and ys :: "'b list"   and n :: "'a"   and i :: "nat" assumes "length xs \<le> length ys"   and "find_first n xs = Some i" shows "fmlookup (fmap_of_list (zip xs ys)) n = Some (ys ! i)"
chagrov: fixes X :: "'a set" obtains "discrete X"   | "\<not> discrete X \<and> part X"   | "\<not> discrete X \<and> ed_ou X"   | "\<not> ed_ou X \<and> open_unresolvable X"   | "\<not> ed_ou X \<and> \<not> part X \<and> extremally_disconnected X"   | "kuratowski X"
nz: shows "fps_exp 1 oo fps_Lambert_W \<noteq> 0"
subresultant'_mat_sylvester_mat: fixes f :: "'b poly"   and g :: "'b poly" shows "(subresultant'_mat 0 0 f g)\<^sup>T = sylvester_mat f g"
nff_\<alpha>_def: fixes dflt :: "'a"   and l :: "'a list"   and i :: "nat" shows "nff_\<alpha> dflt l i \<equiv> if i < length l then l ! i else dflt"
Rep_group_inverse: fixes x :: "'d group" shows "Abs_group (Rep_group x) = x"
srel: shows "is = s2is s \<and> is' = s2is s'"
realistic_executions_def: fixes execs :: "'dom_t \<Rightarrow> 'action_t list list" shows "realistic_executions execs \<equiv> \<forall>d. realistic_execution (execs d)"
mult_zero_right: fixes x :: "V" shows "x * 0 = 0"
prems_3: shows "singleCombinators (ys @ [y])"
has_integral_restrict_UNIV: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and i :: "'b" shows "((\<lambda>x. if x \<in> s then f x else (0::'b)) has_integral i) UNIV = (f has_integral i) s"
A3: shows "A3 \<in> carrier_mat (n - 1) (n - 1)"
top_unique: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and top :: "'b"   and a :: "'b" assumes "class.order_top less_eq less top" shows "less_eq top a = (a = top)"
lexord_Nil_right: fixes x :: "'a list"   and r :: "('a \<times> 'a) set" shows "(x, []) \<notin> lexord r"
image_mset_set_subst_inside: fixes S :: "'b set"   and i :: "'b"   and f :: "'b \<Rightarrow> 'c"   and x :: "'c" assumes "finite S"   and "i \<in> S" shows "image_mset (f(i := x)) (mset_set S) = image_mset f (mset_set S) + {#x#} - {#f i#}"
prv_imp_neg_allI: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and t :: "'trm"   and x :: "'var" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "t \<in> trm"   and "x \<in> var"   and "bprv (imp \<phi> (neg (subst \<chi> t x)))" shows "bprv (imp \<phi> (neg (all x \<chi>)))"
simps_5: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c list \<Rightarrow> 'b"   and x21 :: "'c"   and x22 :: "'c list" shows "(case x21 # x22 of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = f2 x21 x22"
test_compl_de_morgan_2: fixes x :: "'a"   and y :: "'a" shows "tc (x \<cdot> y) = tc x + tc y"
eqfelem_imp_iff: fixes x :: "'c"   and y :: "'c"   and A :: "'c fset" assumes "x = y" shows "(x |\<in>| A) = (y |\<in>| A)"
rel_interior_def: fixes S :: "'a set" shows "rel_interior S = {x. \<exists>T. openin (top_of_set (affine hull S)) T \<and> x \<in> T \<and> T \<subseteq> S}"
Gmult': shows "(*\<star>) \<equiv> aezfun_scalar_mult.Gmult (\<star>)"
A: fixes i :: "'a" assumes "i \<in> J" shows "A i \<in> sets (M i)"
1: fixes x :: "'a"   and y :: "'a" assumes "fst (rep_nat_term x) = fst (rep_nat_term y)"   and "snd (rep_nat_term x) = snd (rep_nat_term y)" shows "x = y"
transfer_rule_numeral_uint: shows "rel_fun (=) cr_uint8 numeral numeral"
exec_gpv_bind: fixes callee :: "'c \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'c) spmf"   and c :: "('f, 'd, 'e) gpv"   and f :: "'f \<Rightarrow> ('b, 'd, 'e) gpv"   and s :: "'c" shows "exec_gpv callee (c \<bind> f) s = exec_gpv callee c s \<bind> (\<lambda>xa. case xa of (x, s') \<Rightarrow> exec_gpv callee (f x) s')"
boundl: fixes x :: "nat list"   and y :: "nat list" assumes "(x, y) \<in> local.Minimal_Solutions"   and "(x, y) \<notin> Special_Solutions" shows "boundl x y"
case_1: fixes f1 :: "'a"   and f2 :: "enat \<Rightarrow> 'a" shows "(case 0 of 0 \<Rightarrow> f1 | eSuc x \<Rightarrow> f2 x) = f1"
step: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream" assumes "ev \<phi> (stl xs)" shows "ev \<phi> xs"
indRelRSTPO_to_SRel_and_TRel_2: fixes P :: "('procS, 'procT) Proc"   and Q :: "('procS, 'procT) Proc"   and SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>R<SRel,TRel> Q" shows "\<forall>SP TQ. SP \<in>S P \<and> TQ \<in>T Q \<longrightarrow> (\<exists>S. (SP, S) \<in> SRel\<^sup>* \<and> (\<lbrakk>S\<rbrakk>, TQ) \<in> TRel\<^sup>*)"
tyenv_sec_eq: fixes mem :: "'Var \<Rightarrow> 'Val"   and mem' :: "'Var \<Rightarrow> 'Val"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and mds :: "Mode \<Rightarrow> 'Var set" assumes "\<forall>x\<in>\<C>. mem x = mem' x"   and "types_wellformed \<Gamma>" shows "(\<forall>x\<in>dom \<Gamma>. x \<notin> mds AsmNoReadOrWrite \<longrightarrow> type_max (the (\<Gamma> x)) mem \<le> dma mem x) = (\<forall>x\<in>dom \<Gamma>. x \<notin> mds AsmNoReadOrWrite \<longrightarrow> type_max (the (\<Gamma> x)) mem' \<le> dma mem' x)"
STEP_REF: fixes x :: "'b"   and x' :: "'a" assumes "(x, x') \<in> R'"   and "b x"   and "b' x'"   and "I x"   and "I' x'" shows "f x \<le> \<Down> R' (f' x')"
add_mset_remove_trivial: fixes x :: "'b"   and M :: "'b multiset" shows "(x, M) - (x, {#}) = M"
union: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c"   and M :: "'c multiset"   and N :: "'c multiset" assumes "comm_monoid_mset f z" shows "comm_monoid_mset.F f z (M + N) = f (comm_monoid_mset.F f z M) (comm_monoid_mset.F f z N)"
finite_cartesian_productD1: fixes A :: "'a set"   and B :: "'b set" assumes "finite (A \<times> B)"   and "B \<noteq> {}" shows "finite A"
L2: shows "Range (r \<inter> A \<times> UNIV) \<in> ce_sets"
uniqAppendElement: fixes l :: "'b list"   and e :: "'b" assumes "uniq l" shows "(e \<notin> set l) = uniq (l @ [e])"
iarray_update_def: fixes a :: "'b iarray"   and i :: "nat"   and x :: "'b" shows "iarray_update a i x = IArray.of_fun (\<lambda>j. if j = i then x else a !! j) (IArray.length a)"
connected_closedin_eq: fixes S :: "'a set" shows "connected S = (\<nexists>E1 E2. closedin (top_of_set S) E1 \<and> closedin (top_of_set S) E2 \<and> E1 \<union> E2 = S \<and> E1 \<inter> E2 = {} \<and> E1 \<noteq> {} \<and> E2 \<noteq> {})"
ncop__ncols: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "\<not> Tarski_neutral_dimensionless.Col Bet A B C \<and> \<not> Tarski_neutral_dimensionless.Col Bet A B D \<and> \<not> Tarski_neutral_dimensionless.Col Bet A C D \<and> \<not> Tarski_neutral_dimensionless.Col Bet B C D"
unpadl_def: fixes m :: "nat"   and v :: "'c vec" shows "unpadl m v \<equiv> vec (dim_vec v - m) (\<lambda>i. v $ (m + i))"
mkArr_eqI: fixes A :: "'s set"   and B :: "'s set"   and F :: "'s \<Rightarrow> 's"   and A' :: "'s set"   and B' :: "'s set"   and F' :: "'s \<Rightarrow> 's" assumes "S.arr (S.mkArr A B F)"   and "A = A'"   and "B = B'"   and "\<And>x. x \<in> A \<Longrightarrow> F x = F' x" shows "S.mkArr A B F = S.mkArr A' B' F'"
clear_subsets: shows "subsets_smash_R (length qs1) (get_subsets_R (calculate_data_R p qs1)) (get_subsets_R (calculate_data_R p qs2)) = get_subsets_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2)))"
graph_plan_neq_mems_state_set_neq_len: fixes PROB :: "(('e, 'f) fmap \<times> ('e, 'g) fmap) set"   and vs :: "'e set" shows "prob_dom (prob_proj PROB vs) = prob_dom PROB \<inter> vs"
strict_implies_not_eq: fixes a :: "'b list"   and b :: "'b list" assumes "a <p b" shows "a \<noteq> b"
ty_dom_vrs_prm_simp: fixes pi :: "(vrs \<times> vrs) list"   and \<Gamma> :: "binding list" shows "ty_dom (pi \<bullet> \<Gamma>) = ty_dom \<Gamma>"
assms_2: shows "\<Theta>,\<Gamma> \<turnstile> mk_eq f g"
canon_ang_eqI: fixes \<alpha>' :: "real"   and \<alpha> :: "real" assumes "\<exists>x. \<alpha>' - \<alpha> = 2 * real_of_int x * pi"   and "- pi < \<alpha>' \<and> \<alpha>' \<le> pi" shows "\<downharpoonright>\<alpha>\<downharpoonleft> = \<alpha>'"
ext_is_act: fixes a :: "'d"   and S :: "'d set \<times> 'd set \<times> 'd set" assumes "a \<in> externals S" shows "a \<in> actions S"
mono_det_FD_onside: fixes P :: "'a process"   and P' :: "'a process"   and S :: "'a process" assumes "P \<le> P'" shows "(P [+] S) \<le> (P' [+] S)"
skl1_inv6I: fixes x :: "skl1_state" assumes "\<And>A B Rb Ra gnx Ni. \<lbrakk>guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr>; in_progressS (progress x Ra) {xnx, xni, xnr, xgnx, xgny, xsk, xEnd}; sklvl1.guessed_frame Ra xgnx = Some gnx; sklvl1.guessed_frame Ra xni = Some Ni; sklvl1.guessed_frame Ra xgny = Some (Exp Gen (NonceF (Rb $ ny))); sklvl1.guessed_frame Ra xnr = Some (NonceF (Rb $ nr)); guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr>; progress x Rb = Some {xny, xni, xnr, xgnx, xgny, xsk}; can_signal x A B\<rbrakk> \<Longrightarrow> signalsResp x (Commit A B \<langle>Ni, NonceF (Rb $ nr), Exp gnx (NonceF (Rb $ ny))\<rangle>) < signalsResp x (Running A B \<langle>Ni, NonceF (Rb $ nr), Exp gnx (NonceF (Rb $ ny))\<rangle>)" shows "x \<in> skl1_inv6"
sgn_sign_eq: fixes x :: "'a" shows "sgn x = of_int (sign x)"
sum_zero_eq_neg: fixes x :: "int \<times> int"   and y :: "int \<times> int" assumes "x \<in> carrier lucas_lehmer_ring"   and "y \<in> carrier lucas_lehmer_ring"   and "x \<oplus>\<^bsub>lucas_lehmer_ring\<^esub> y = \<zero>\<^bsub>lucas_lehmer_ring\<^esub>" shows "x = \<ominus>\<^bsub>lucas_lehmer_ring\<^esub> y"
A: shows "Pi\<^sub>M (insert (Suc v) (Suc ` V)) (case_nat M N) = Pi\<^sub>M (insert (Suc v) (Suc ` V)) (case_nat (count_space {}) N)"
init_Power_dfa: shows "dfa.init Power_dfa = HF (epsclo (nfa.init M))"
ex_orthogonal_current: fixes \<Gamma> :: "('a, 'b) web_scheme" assumes "bounded_countable_web \<Gamma>" shows "\<exists>f S. web_flow \<Gamma> f \<and> separating \<Gamma> S \<and> orthogonal_current \<Gamma> f S"
predicate1D: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool"   and x :: "'c" assumes "P \<le> Q"   and "P x" shows "Q x"
WT_callee_of_core: fixes \<I>_adv :: "('a, 'b) \<I>"   and \<I>_usr :: "('c, 'd) \<I>"   and I :: "'e \<Rightarrow> bool"   and core :: "('e, 'f, 'a, 'c, 'b, 'd) core"   and s :: "'e" assumes "WT_core \<I>_adv \<I>_usr I core"   and "I s" shows "\<I>_full \<oplus>\<^sub>\<I> (\<I>_adv \<oplus>\<^sub>\<I> \<I>_usr) \<turnstile>c callee_of_core core s \<surd>"
obt1: shows "e \<in> outgoing t \<and> f e \<noteq> (0::'capacity)"