hcju/isabelleps huggingface dataset:
elims: fixes x :: "trilean"   and xa :: "trilean"   and y :: "trilean" assumes "x \<and>? xa = y" obtains uu_ :: "trilean" where "x = uu_" and "xa = invalid" and "y = invalid"   | "x = invalid" and "xa = true" and "y = invalid"   | "x = invalid" and "xa = false" and "y = invalid"   | "x = true" and "xa = true" and "y = true"   | "x = true" and "xa = false" and "y = false"   | "x = false" and "xa = false" and "y = false"   | "x = false" and "xa = true" and "y = false"
inline_def: fixes c :: "('b, 'call, 'ret) gpv"   and s :: "'s" shows "local.inline c s = local.inline_aux (Inl (c, s))"
lincomb': shows "(\<bullet>\<star>) \<equiv> scalar_mult.lincomb (\<star>)"
strategyproofness: shows "(pmf (sds R17) c + pmf (sds R17) a < pmf (sds R5) c + pmf (sds R5) a \<or> pmf (sds R17) c + (pmf (sds R17) a + pmf (sds R17) d) < pmf (sds R5) c + (pmf (sds R5) a + pmf (sds R5) d)) \<or> pmf (sds R17) c + pmf (sds R17) a = pmf (sds R5) c + pmf (sds R5) a \<and> pmf (sds R17) c + (pmf (sds R17) a + pmf (sds R17) d) = pmf (sds R5) c + (pmf (sds R5) a + pmf (sds R5) d)"
pcorres_def: fixes \<phi> :: "'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> bool"   and A :: "bool \<Rightarrow> ('d \<Rightarrow> real) \<Rightarrow> 'd \<Rightarrow> real"   and B :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "pcorres \<phi> G A B = (\<forall>Q. unitary Q \<longrightarrow> \<guillemotleft> G \<guillemotright> && (wp A Q \<circ> \<phi>) = \<guillemotleft> G \<guillemotright> && wp B (Q \<circ> \<phi>))"
sign_simps_18: fixes aa :: "'c"   and ba :: "'c"   and ca :: "'c" shows "aa * (ba + ca) = aa * ba + aa * ca"
equality: fixes r :: "('d, 'c, 'e) prism_scheme"   and r' :: "('d, 'c, 'e) prism_scheme" assumes "match\<^bsub>r\<^esub> = match\<^bsub>r'\<^esub>"   and "build\<^bsub>r\<^esub> = build\<^bsub>r'\<^esub>"   and "prism.more r = prism.more r'" shows "r = r'"
p_def: fixes x :: "nat" shows "p x = c_pair x x"
prognosis_called: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and as :: "Arity list"   and a :: "Arity"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and x :: "Vars.var"   and S :: "stack_elem list" shows "once \<sqsubseteq> prognosis ae as a (\<Gamma>, Var x, S) x"
rec_actn_def: shows "rec_actn = Cn 3 rec_add [Cn 3 rec_mult [Cn 3 rec_entry [recf.id 3 0, Cn 3 rec_add [Cn 3 rec_mult [Cn 3 (constn 4) [recf.id 3 0], Cn 3 rec_minus [recf.id 3 1, Cn 3 (constn 1) [recf.id 3 0]]], Cn 3 rec_mult [Cn 3 (constn 2) [recf.id 3 0], Cn 3 rec_scan [recf.id 3 2]]]], Cn 3 rec_noteq [recf.id 3 1, Cn 3 (constn 0) [recf.id 3 0]]], Cn 3 rec_mult [Cn 3 (constn 4) [recf.id 3 0], Cn 3 rec_eq [recf.id 3 1, Cn 3 (constn 0) [recf.id 3 0]]]]"
elem_in_group_sizes_min_of_size: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and groups :: "'b set set"   and a :: "nat" assumes "group_design point_set block_collection groups"   and "a \<in> group_design.group_sizes groups" shows "1 \<le> group_design.groups_of_size groups a"
add_delete_edge: fixes a :: "'v"   and w :: "'w"   and c :: "'v" assumes "(a, w, c) \<in> E" shows "add_edge a w c (delete_edge a w c G) = G"
set_of_subset_iff: fixes X :: "'a interval"   and Y :: "'a interval" shows "(set_of X \<subseteq> set_of Y) = (lower Y \<le> lower X \<and> upper X \<le> upper Y)"
atMost_iff: fixes i :: "'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and k :: "'e" shows "(i \<in> ord.atMost less_eq k) = less_eq i k"
tra: shows "a' \<otimes> b \<sim> a \<otimes> b"
refl: fixes x :: "'a\<^sub>m" assumes "x \<in> UNIV" shows "bo x x"
word_add_not: fixes x :: "'a Word.word" shows "x + ~~ x = - 1"
fold_map_of: fixes xs :: "'g list"   and k :: "nat"   and m :: "'g \<Rightarrow> nat option" assumes "distinct xs" shows "fold (\<lambda>x (k, m). (Suc k, m(x \<mapsto> k))) xs (k, m) = (k + length xs, m ++ map_of (xs || [k..<k + length xs]))"
assms_3: shows "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"
subset_open_segment: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" shows "(open_segment a b \<subseteq> open_segment c d) = (a = b \<or> a \<in> closed_segment c d \<and> b \<in> closed_segment c d)"
run_put_state: fixes s :: "'b"   and m :: "('b, 'a) stateT"   and s' :: "'b" shows "run_state (put_state s m) s' = run_state m s"
extremum_unique: fixes a :: "'a list" shows "suffix a [] = (a = [])"
finite_adjacentset: fixes C :: "'a set" assumes "chamber C" shows "finite (adjacentset C)"
OUren: fixes ODE :: "('sf, 'sz) ODE"   and I :: "('sf, 'b, 'sz) interp"   and x :: "'sz"   and y :: "'sz"   and \<nu> :: "(real, 'sz) vec" assumes "ORadmit ODE" shows "ODE_sem I (OUrename x y ODE) \<nu> = RSadj x y (ODE_sem I ODE (RSadj x y \<nu>))"
finite_measure_finite_Union: fixes S :: "'c set"   and A :: "'c \<Rightarrow> 'key set" assumes "finite S"   and "A ` S \<subseteq> K.events"   and "disjoint_family_on A S" shows "K.prob (\<Union> (A ` S)) = (\<Sum>i\<in>S. K.prob (A i))"
same_length_different: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs \<noteq> ys"   and "length xs = length ys" shows "\<exists>pre x xs' y ys'. x \<noteq> y \<and> xs = pre @ [x] @ xs' \<and> ys = pre @ [y] @ ys'"
path_inv_Leaf: fixes p :: "bool list" shows "path p Leaf = (p = [])"
implem_Tags_aux: fixes x :: "tag"   and A :: "agent"   and B :: "agent"   and M :: "msg" shows "implem (Chan x A B M) \<notin> Tags"
deg_shifts_superset: fixes fs :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) list"   and d :: "nat" shows "set fs \<subseteq> set (deg_shifts d fs)"
a_starE: fixes M :: "trm"   and M' :: "trm" assumes "M \<longrightarrow>\<^sub>a* M'" shows "M = M' \<or> (\<exists>N. M \<longrightarrow>\<^sub>a N \<and> N \<longrightarrow>\<^sub>a* M')"
assms_2: fixes a :: "'a set" assumes "a \<subseteq> A" shows "finite (f a)"
max_eqR: fixes a :: "'b"   and b :: "'b" assumes "a \<le> b" shows "max a b = b"
less_le_TC_trans: fixes x :: "V"   and y :: "V"   and z :: "V" assumes "x \<sqsubset> y"   and "y \<sqsubseteq> z" shows "x \<sqsubset> z"
coincidence_frechet': fixes \<theta> :: "('d, 'e) trm"   and \<nu> :: "(real, 'e) vec \<times> (real, 'e) vec"   and \<nu>' :: "(real, 'e) vec \<times> (real, 'e) vec"   and I :: "('d, 'f, 'e) interp"   and J :: "('d, 'f, 'e) interp" assumes "dfree \<theta>"   and "Vagree \<nu> \<nu>' (FVDiff \<theta>)"   and "Iagree I J {Inl x |x. x \<in> SIGT \<theta>}" shows "frechet I \<theta> (fst \<nu>) (snd \<nu>) = frechet J \<theta> (fst \<nu>') (snd \<nu>')"
simps_2: fixes \<tau> :: "ty"   and \<sigma> :: "ty" shows "RED (\<tau> \<rightarrow> \<sigma>) = {t. \<forall>u. u \<in> RED \<tau> \<longrightarrow> App t u \<in> RED \<sigma>}"
oalist_inv_raw_ConsI: fixes xs :: "('a \<times> 'e) list"   and v :: "'e"   and k :: "'a" assumes "oalist_inv_raw xs"   and "v \<noteq> (0::'e)"   and "xs \<noteq> [] \<Longrightarrow> lt k (fst (hd xs))" shows "oalist_inv_raw ((k, v) # xs)"
connected_space_def: fixes X :: "'b topology" shows "connected_space X \<equiv> \<nexists>E1 E2. openin X E1 \<and> openin X E2 \<and> topspace X \<subseteq> E1 \<union> E2 \<and> E1 \<inter> E2 = {} \<and> E1 \<noteq> {} \<and> E2 \<noteq> {}"
kcomp_eq: fixes f :: "'c \<Rightarrow> 'd set"   and g :: "'d \<Rightarrow> 'b set"   and x :: "'c" shows "(f ; g) x = \<mu> {g y |y. y \<in> f x}"
inverse_arrowsE: fixes f :: "'a"   and g :: "'a" assumes "C.inverse_arrows f g" obtains "C.ide (g \<cdot>\<^sub>C f)" and "C.ide (f \<cdot>\<^sub>C g)"
cases: fixes a1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and a2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "\<tau>mred1 a1 a2" obtains s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" and tl :: "'t \<times> ('l, 't, 'x1, 'm1, 'w, 'o) thread_action" where "a1 = s" and "a2 = s'" and "r1.redT s tl s'" and "m\<tau>move1 s tl s'"
lower_closure: shows "\<pi>\<^sup>* \<in> carrier \<X> \<rightarrow> carrier \<Y>"
finite_PiE_iff: fixes I :: "'f set"   and S :: "'f \<Rightarrow> 'g set" shows "finite (Pi\<^sub>E I S) = (Pi\<^sub>E I S = {} \<or> finite {i \<in> I. \<nexists>a. S i \<subseteq> {a}} \<and> (\<forall>i\<in>I. finite (S i)))"
set_empty: fixes xs :: "'c list" shows "(set xs = {}) = (xs = [])"
samp_uni_add_mult: fixes x :: "nat"   and q :: "nat"   and x' :: "nat"   and y' :: "nat"   and y :: "nat" assumes "coprime x q"   and "x' < q"   and "y' < q"   and "(y + x * x') mod q = (y + x * y') mod q" shows "x' = y'"
Rep_cl_op_im_inverse: fixes x :: "'b cl_op_im" shows "Abs_cl_op_im (Rep_cl_op_im x) = x"
sum_Un_introL: fixes A1 :: "'a set"   and B1 :: "'a set"   and C1 :: "'a set"   and x :: "'b"   and x1 :: "'b"   and x2 :: "'b"   and f1 :: "'a \<Rightarrow> 'b" assumes "A1 = B1 \<union> C1"   and "x = x1 + x2"   and "finite A1"   and "B1 \<inter> C1 = {}"   and "sum f1 B1 = x1"   and "sum f1 C1 = x2" shows "sum f1 A1 = x"
simps: fixes a1 :: "string_binop"   and a2 :: "'a OCL_Types.type"   and a3 :: "'b OCL_Types.type"   and a4 :: "'c OCL_Types.type" shows "string_binop_type a1 a2 a3 a4 = (a1 = ConcatOp \<and> a2 = String[1] \<and> a3 = String[1] \<and> a4 = String[1] \<or> a1 = EqualsIgnoreCaseOp \<and> a2 = String[1] \<and> a3 = String[1] \<and> a4 = Boolean[1] \<or> a1 = string_binop.LessOp \<and> a2 = String[1] \<and> a3 = String[1] \<and> a4 = Boolean[1] \<or> a1 = string_binop.LessEqOp \<and> a2 = String[1] \<and> a3 = String[1] \<and> a4 = Boolean[1] \<or> a1 = string_binop.GreaterOp \<and> a2 = String[1] \<and> a3 = String[1] \<and> a4 = Boolean[1] \<or> a1 = string_binop.GreaterEqOp \<and> a2 = String[1] \<and> a3 = String[1] \<and> a4 = Boolean[1] \<or> a1 = IndexOfOp \<and> a2 = String[1] \<and> a3 = String[1] \<and> a4 = Integer[1] \<or> (\<exists>\<tau>. a1 = AtOp \<and> a2 = String[1] \<and> a3 = \<tau> \<and> a4 = String[1] \<and> \<tau> = UnlimitedNatural[1]\<midarrow>Integer[1]))"
col_space_def: fixes A :: "(('a, 'c) vec, 'b) vec" shows "col_space A = vec.span (columns A)"
Identity: fixes \<Psi> :: "'b" shows "\<Psi> \<otimes> \<one> \<simeq> \<Psi>"
injective_mult_closed: fixes x :: "'a"   and y :: "'a" assumes "injective x"   and "injective y" shows "injective (x * y)"
bn_abs_residual_fresh: fixes \<alpha> :: "'a"   and P :: "'b" shows "bn \<alpha> \<sharp>* \<langle>\<alpha>,P\<rangle>"
conjugation_pair_multl: fixes y :: "'a" shows "conjugation_pair (\<lambda>x. x \<cdot> y) (\<lambda>x. x \<lhd> y)"
RRR_interior: shows "interior RRR = {0<..<1} \<union> {1<..<2}"
true_conj_zero_1: fixes P :: "'b upred" shows "(P \<and> false) = false"
antimono_pow2: shows "decseq (\<lambda>i. (f ^^ i) (f top))"
assms_3: shows "diff_invariant I (\<lambda>t. f) (\<lambda>s. {t. 0 \<le> t}) UNIV 0 G"
triangle_in_hom_1: shows "\<guillemotleft>(\<epsilon>' \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g', f] \<cdot> (f \<star> \<eta>') : f \<star> src f \<Rightarrow> trg f \<star> f\<guillemotright>"
emeasure_positive: fixes M :: "'d measure" shows "positive (sets M) (emeasure M)"
map_2: fixes f :: "'f \<Rightarrow> 'e"   and x2 :: "'f" shows "map_option f (Some x2) = Some (f x2)"
findMin_def: fixes bq :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewBinomialTree list" shows "BsSkewBinomialHeapStruc.findMin bq = BsSkewBinomialHeapStruc.val (BsSkewBinomialHeapStruc.getMinTree bq)"
mirror_unfold: fixes x :: "'b"   and l :: "'b tree"   and r :: "'b tree" shows "mirror (Node x l r) = Node x (mirror r) (mirror l)"
smap_alt: fixes f :: "'d \<Rightarrow> 'b"   and s :: "'d stream"   and s' :: "'b stream" shows "(smap f s = s') = (\<forall>n. f (s !! n) = s' !! n)"
simps_3: fixes p :: "real \<times> real \<times> real" shows "substNegInfinityUni (LeqUni p) = OrUni (alternateNegInfinity' p) (allZero' p)"
smult_one: fixes c :: "'a" shows "Polynomial.smult c 1 = [:c:]"
openin_clauses_2: fixes U :: "'c topology"   and S :: "'c set"   and T :: "'c set" assumes "openin U S"   and "openin U T" shows "openin U (S \<inter> T)"
lm015: fixes P :: "('c \<times> 'd) set"   and X :: "'c set"   and Y :: "'c set" shows "P || (X \<inter> Y) \<subseteq> P || X \<and> P outside (X \<union> Y) \<subseteq> P outside X"
inter_def: fixes l :: "Lines"   and m :: "Lines" shows "Pascal_Property.inter l m \<equiv> SOME P. P \<in> inters l m"
addr_loc_type_def: fixes P :: "'a prog" shows "addr_loc_type P = h.addr_loc_type P undefined"
is_aligned_0: fixes n :: "nat" shows "is_aligned 0 n"
sums_iff_shift': fixes n :: "nat"   and s :: "'a" shows "(\<lambda>i. f (i + n)) sums (s - sum f {..<n}) = f sums s"
has_derivative_diff: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and g' :: "'a \<Rightarrow> 'b" assumes "(f has_derivative f') F"   and "(g has_derivative g') F" shows "((\<lambda>x. f x - g x) has_derivative (\<lambda>x. f' x - g' x)) F"
scaleR_le_cancel_left_pos: fixes c :: "real"   and a :: "'a"   and b :: "'a" assumes "0 < c" shows "(c *\<^sub>R a \<le> c *\<^sub>R b) = (a \<le> b)"
cfun_map_beta: fixes a :: "'b \<rightarrow> 'c"   and b :: "'d \<rightarrow> 'a"   and f :: "'c \<rightarrow> 'd"   and x :: "'b" shows "cfun_map\<cdot>a\<cdot>b\<cdot>f\<cdot>x = b\<cdot>(f\<cdot>(a\<cdot>x))"
assms: shows "\<forall>x. Substable cond (\<psi> x)"
set_mrbst_diff: fixes t' :: "'a tree"   and t1 :: "'a tree"   and t2 :: "'a tree" assumes "t' \<in> set_pmf (mrbst_diff t1 t2)"   and "bst t1"   and "bst t2" shows "set_tree t' = set_tree t1 - set_tree t2"
path_state_closure: fixes s :: "'s"   and \<tau> :: "'e list"   and s' :: "'s" assumes "s \<tau>\<Longrightarrow>\<^bsub>SES\<^esub> s'"   and "s \<in> S\<^bsub>SES\<^esub>" shows "s' \<in> S\<^bsub>SES\<^esub>"
ahm_empty_const_def: shows "ahm_empty_const \<equiv> HashMap (ArrayHashMap_Impl.ahm_empty ())"
sorted_trans: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "xs < ys"   and "ys < zs"   and "sorted ys"   and "ys \<noteq> []" shows "xs < zs"
one_im_sheaf_def: shows "idim.one_im_sheaf \<equiv> \<lambda>V. \<one>\<^bsub>identity S \<^sup>\<inverse> S V\<^esub>"
isos_compose: fixes f :: "'a"   and f' :: "'a" assumes "L'.C.iso f"   and "L'.C.iso f'"   and "L'.C.seq f' f" shows "L'.C.iso (local.Left a' f' f)"
tendsto_iff_Liminf_eq_Limsup: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'd"   and f0 :: "'d" assumes "F \<noteq> bot" shows "(f \<longlongrightarrow> f0) F = (Liminf F f = f0 \<and> Limsup F f = f0)"
ahm_delete_def: fixes k :: "'a"   and hm :: "('a, 'b) hashmap" shows "ahm_delete k hm = HashMap (ArrayHashMap_Impl.ahm_delete k (impl_of hm))"
cases: fixes r :: "sig" obtains name :: "mname" and parTs :: "ty list" where "r = \<lparr>name = name, parTs = parTs\<rparr>"
small_these_tiny_tdghms: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<GG> :: "V" shows "small {\<NN>. \<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<BB>}"
tuple_simps_3: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "span f g" shows "cod (hfsetcat.tuple f g) = hfsetcat.prod (cod f) (cod g)"
ifTrue_deterministic: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and r :: "'a"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<tau> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<E> :: "('a, 'b, 'c) cntxt"   and e1 :: "('a, 'b, 'c) expr"   and e2 :: "('a, 'b, 'c) expr"   and s' :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" assumes "substitution subst"   and "s r = Some (\<sigma>, \<tau>, \<E> [Ite (VE (CV T)) e1 e2])" shows "substitution.revision_step subst r s s' = (s' = s(r \<mapsto> (\<sigma>, \<tau>, \<E> [e1])))"
less_eq_hmset_zhmsetE: fixes M :: "zhmultiset"   and N :: "zhmultiset" assumes "M \<le> N" obtains A :: "hmultiset" and B :: "hmultiset" and C :: "zhmultiset" where "M = zhmset_of A + C" and "N = zhmset_of B + C" and "A \<le> B"
binrelchain_append_reduce2: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list" assumes "binrelchain P (xs @ ys)" shows "binrelchain P ys"
mono: fixes guard :: "'a \<Rightarrow> bool"   and x :: "'a"   and body :: "'a \<Rightarrow> 'a spmf" shows "spmf.mono_body (\<lambda>while. if guard x then body x \<bind> while else return_spmf x)"
mem_lt_imp_MIN_lt: fixes s :: "nat set"   and k :: "nat" assumes "\<exists>x. x \<in> s \<and> x < k" shows "Inf s < k"
simps_10: fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term" shows "\<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[t, u, v\<^bold>]\<rbrace> = \<alpha>' (\<lbrace>t\<rbrace>, \<lbrace>u\<rbrace>, \<lbrace>v\<rbrace>)"
sort_rev_eq_sort: fixes xs :: "'d list" assumes "distinct xs" shows "sort (rev xs) = sort xs"
is_non_overlapping_def: fixes P :: "'c set set" shows "is_non_overlapping P = (\<forall>X\<in>P. \<forall>Y\<in>P. (X \<inter> Y \<noteq> {}) = (X = Y))"
mem_carrier: fixes H :: "'a set"   and G :: "('a, 'b) monoid_scheme"   and x :: "'a" assumes "subgroup H G"   and "x \<in> H" shows "x \<in> carrier G"
hequiv_proj: fixes A :: "'c set"   and H :: "(('d, 'c) fm list \<times> 'c) set"   and i :: "'c"   and a :: "'c"   and b :: "'c" assumes "Hintikka A H"   and "Nom a at i in' H"   and "a \<in> A"   and "Nom b at i in' H"   and "b \<in> A" shows "proj (hequiv_rel A H) a = proj (hequiv_rel A H) b"
inter_set_filter: fixes A :: "'a set"   and xs :: "'a list" shows "A \<inter> set xs = set (project A xs)"
set_plus_mono: fixes C :: "'a set"   and D :: "'a set"   and a :: "'a" assumes "C \<subseteq> D" shows "a +o C \<subseteq> a +o D"
typ_of_eq_ax_4: shows "typ_of eq_intr_ax = Some propT"
homotopic_with_compose_continuous_map_left: fixes p :: "('b \<Rightarrow> 'c) \<Rightarrow> bool"   and X1 :: "'b topology"   and X2 :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and X3 :: "'d topology"   and h :: "'c \<Rightarrow> 'd"   and q :: "('b \<Rightarrow> 'd) \<Rightarrow> bool" assumes "homotopic_with p X1 X2 f g"   and "continuous_map X2 X3 h"   and "\<And>j. p j \<Longrightarrow> q (h \<circ> j)" shows "homotopic_with q X1 X3 (h \<circ> f) (h \<circ> g)"
mono: shows "SecurityInvariant_withOffendingFlows.sinvar_mono sinvar_spec"
par_col4__par: fixes E :: "'p"   and F :: "'p"   and G :: "'p"   and H :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "E \<noteq> F"   and "G \<noteq> H"   and "A B Par C D"   and "Col A B E"   and "Col A B F"   and "Col C D G"   and "Col C D H" shows "E F Par G H"
x: shows "Derives1 u (fst d) (snd d) x \<and> Derivation x D v"
powers_subdistl: fixes x :: "'a"   and y :: "'a" shows "(\<Squnion>i. x \<cdot> y ^ i) \<le> x \<cdot> qstar y"
per_root_spref_powE: fixes u :: "'b list"   and r :: "'b list" assumes "u \<le>p r\<^sup>\<omega>" obtains k :: "nat" where "u <p r \<^sup>@ k"
root_1: assumes "p \<noteq> 0" shows "card (proots_within p B) = card (proots_within (p \<circ>\<^sub>p q) A)"
diff_sum_integral_has_integral_int: shows "((\<lambda>t. (frac t - 1 / 2) *\<^sub>R f' t) has_integral S - I - (f (real_of_int b) - f (real_of_int a)) /\<^sub>R 2) {real_of_int a..real_of_int b}"
dvd_mod: fixes k :: "'a"   and m :: "'a"   and n :: "'a" assumes "k dvd m"   and "k dvd n" shows "k dvd m mod n"
intro: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e" assumes "category A"   and "category B"   and "functor A B F"   and "functor A B G"   and "natural_isomorphism A B F G \<tau>" shows "inverse_transformation A B F G \<tau>"
A_def: shows "A = (\<lambda>n. P n -` {True} \<inter> space M - (\<Union>m<n. P m -` {True} \<inter> space M))"
box': fixes s :: "real" assumes "s \<in> {0..t}" shows "directional_derivative I (f1 fid2 vid1) (fst (mk_v I (OVar vid1) (sol 0, b) (sol s)), snd (mk_v I (OVar vid1) (sol 0, b) (sol s))) \<le> directional_derivative I (f1 fid1 vid1) (fst (mk_v I (OVar vid1) (sol 0, b) (sol s)), snd (mk_v I (OVar vid1) (sol 0, b) (sol s)))"
uniq_zero: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c"   and p\<^sub>1 :: "'c point"   and p\<^sub>2 :: "'c point" assumes "ell_field R"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "ell_field.nonsingular R a b"   and "ell_field.on_curve R a b p\<^sub>1"   and "ell_field.on_curve R a b p\<^sub>2"   and "ell_field.add R a p\<^sub>1 p\<^sub>2 = p\<^sub>2" shows "p\<^sub>1 = Infinity"
1: fixes x' :: "'a"   and y' :: "'a"   and z :: "'a" assumes "x' \<in> I +> x"   and "y' \<in> I +> y"   and "z \<in> I +> x' \<otimes> y'" shows "z \<in> I +> x \<otimes> y"
Icc_subset_Ici_iff: fixes l :: "'a"   and h :: "'a"   and l' :: "'a" shows "({l..h} \<subseteq> {l'..}) = (\<not> l \<le> h \<or> l' \<le> l)"
op_norm_ge_0: fixes A :: "(('b, 'c) vec, 'd) vec" shows "0 \<le> \<parallel>A\<parallel>\<^sub>o\<^sub>p"
shift_then_mask_eq_shift_low_bits: fixes x :: "'b word"   and low_bits :: "nat"   and high_bits :: "nat" assumes "x \<le> mask (low_bits + high_bits)" shows "(x >> low_bits) AND mask high_bits = x >> low_bits"
foldr_conj_map: fixes bs :: "real list"   and f :: "'a \<Rightarrow> fm"   and xs :: "'a list" shows "Ifm bs (foldr MIR.conj (map f xs) T) = (\<forall>x\<in>set xs. Ifm bs (f x))"
distinct_56: fixes x81 :: "nat"   and x82 :: "msg"   and x71 :: "msg"   and x72 :: "msg" shows "Crypt x81 x82 \<noteq> \<lbrace>x71, x72\<rbrace>"
sgn_eq_0_iff: fixes abs :: "'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "(sgn a = zero) = (a = zero)"
simps_26: fixes f1 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> k \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f3 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat \<Rightarrow> 'a"   and f4 :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and x11 :: "nat"   and x12 :: "nat"   and x13 :: "nat"   and x14 :: "nat"   and x15 :: "k"   and x16 :: "nat"   and x17 :: "nat"   and x18 :: "nat" shows "(case Rreq x11 x12 x13 x14 x15 x16 x17 x18 of Rreq x xa xb xc xd xe xf xg \<Rightarrow> f1 x xa xb xc xd xe xf xg | Rrep x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd | Rerr x xa \<Rightarrow> f3 x xa | Newpkt x xa \<Rightarrow> f4 x xa | Pkt x xa xb \<Rightarrow> f5 x xa xb) = f1 x11 x12 x13 x14 x15 x16 x17 x18"
fst_eqD: fixes x :: "'e"   and y :: "'f"   and a :: "'e" assumes "fst (x, y) = a" shows "x = a"
distinct_1: fixes x1 :: "'a" shows "Action x1 \<noteq> NoAction"
not_eq_order_implies_strict: fixes a :: "'c word"   and b :: "'c word" assumes "a \<noteq> b"   and "a \<le>s b" shows "a <s b"
take_all_iff: fixes n :: "nat"   and xs :: "'a list" shows "(take n xs = xs) = (length xs \<le> n)"
update_zip: fixes xs :: "'b list"   and ys :: "'c list"   and i :: "nat"   and xy :: "'b \<times> 'c" shows "(zip xs ys)[i := xy] = zip (xs[i := fst xy]) (ys[i := snd xy])"
chine_simps_2: shows "C.dom g.chine = g.apex"