hcju/isabelleps huggingface dataset:
remove_child_heap_is_wellformed_preserved_1: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and child :: "(_) node_ptr"   and h' :: "(_) heap" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "h \<turnstile> remove_child ptr child \<rightarrow>\<^sub>h h'"   and "known_ptrs h"   and "type_wf h" shows "type_wf h'"
llength_llist_of: fixes xs :: "'c list" shows "llength (llist_of xs) = enat (length xs)"
unit_imp_mod_eq_0: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and b :: "'b"   and a :: "'b" assumes "class.euclidean_semiring divide plus minus zero times one modulo euclidean_size"   and "dvd.dvd times b one" shows "modulo a b = zero"
cup_pmatch_list_typerr: fixes vs :: "v list"   and ps :: "pat list"   and cenv :: "(char list, char list, nat \<times> tid_or_exn) namespace" assumes "length vs = length ps" shows "Matching.fold2 (\<lambda>p v m. case m of No_match \<Rightarrow> No_match | Match_type_error \<Rightarrow> Match_type_error | Match env \<Rightarrow> cupcake_pmatch cenv p v env) Match_type_error ps vs Match_type_error = Match_type_error"
simps_1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "CMP_BRANCH LT x y z = x"
smap_union_comm: fixes m1 :: "'a \<Rightarrow> 'b set"   and m2 :: "'a \<Rightarrow> 'b set" shows "m1 \<union>. m2 = m2 \<union>. m1"
simps_2: fixes n :: "nat"   and p :: "nat"   and ps :: "nat list" shows "totient_aux1 n (p # ps) = totient_aux1 (n - n div p) ps"
guarantees_JN_I: fixes i :: "'c"   and I :: "'c set"   and F :: "'c \<Rightarrow> 'd program"   and X :: "'d program set"   and Y :: "'d program set" assumes "i \<in> I"   and "F i \<in> X guarantees Y"   and "OK I F" shows "JOIN I F \<in> X guarantees Y"
map_filter_map_option_const: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'c option"   and xs :: "'b list" shows "List.map_filter (\<lambda>x. map_option (\<lambda>y. f x) (g (f x))) xs = filter (\<lambda>x. g x \<noteq> None) (map f xs)"
P_def: shows "P \<equiv> \<lambda>\<delta>. subst_domain \<delta> = set X \<and> ground (subst_range \<delta>)"
no_imp_one: fixes x :: "'a"   and p :: "'a list" assumes "no_call_in_path x p" shows "one_call_in_path x p"
greaterThan_Suc: fixes k :: "nat" shows "{Suc k<..} = {k<..} - {Suc k}"
type_copy_map_comp0_undo: fixes Rep :: "'c \<Rightarrow> 'd"   and Abs :: "'d \<Rightarrow> 'c"   and Rep' :: "'e \<Rightarrow> 'f"   and Abs' :: "'f \<Rightarrow> 'e"   and Rep'' :: "'g \<Rightarrow> 'h"   and Abs'' :: "'h \<Rightarrow> 'g"   and M :: "'h \<Rightarrow> 'f"   and M1 :: "'d \<Rightarrow> 'f"   and M2 :: "'h \<Rightarrow> 'd" assumes "type_definition Rep Abs UNIV"   and "type_definition Rep' Abs' UNIV"   and "type_definition Rep'' Abs'' UNIV"   and "Abs' \<circ> M \<circ> Rep'' = Abs' \<circ> M1 \<circ> Rep \<circ> (Abs \<circ> M2 \<circ> Rep'')" shows "M1 \<circ> M2 = M"
intros_2: fixes \<sigma> :: "'var \<Rightarrow> 'val"   and k :: "nat"   and \<sigma>' :: "'var \<Rightarrow> 'val"   and k' :: "nat"   and n :: "nat"   and v :: "'var"   and n' :: "nat" assumes "((\<sigma>, k), \<sigma>', k') \<in> cp"   and "n dd\<^bsup>path \<sigma>,v\<^esup>\<rightarrow> k"   and "n' dd\<^bsup>path \<sigma>',v\<^esup>\<rightarrow> k'"   and "cs\<^bsup>path \<sigma>\<^esup> n = cs\<^bsup>path \<sigma>'\<^esup> n'"   and "(\<sigma>\<^bsup>n\<^esup>) v \<noteq> (\<sigma>'\<^bsup>n'\<^esup>) v" shows "((\<sigma>, n), \<sigma>', n') \<in> cp"
PiM_eqI: fixes I :: "'i set"   and P :: "('i \<Rightarrow> 'a) measure" assumes "finite I"   and "sets P = sets (Pi\<^sub>M I M)"   and "\<And>A. (\<And>i. i \<in> I \<Longrightarrow> A i \<in> events i) \<Longrightarrow> emeasure P (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))" shows "P = Pi\<^sub>M I M"
Pxy_nn: fixes x :: "'d"   and y :: "'c" assumes "x \<in> space S"   and "y \<in> space T" shows "0 \<le> Pxy (x, y)"
non_null_1: fixes i :: "'b" assumes "i \<in> S" shows "0 < a i"
assms_3: shows "angle b1 a1 c1 = angle b2 a2 c2"
secure_def: fixes p :: "L"   and G :: "Var \<Rightarrow> L"   and c :: "IMP"   and H :: "Var \<Rightarrow> L" shows "secure p G c H = (\<forall>q. ARSsecure (Q p H) (EQ G q) (EQ H q) c)"
Texp_App: fixes x :: "Vars.var"   and e :: "exp"   and a :: "Arity" shows "many_calls x \<otimes>\<otimes> Texp e\<cdot>(inc\<cdot>a) \<sqsubseteq> Texp (App e x)\<cdot>a"
fresh_fresh_rename_idns: fixes G :: "variable set"   and n :: "nat"   and B :: "Core.term"   and insts :: "((variable \<times> typ) \<times> Core.term) list" assumes "finite G" shows "\<forall>nm\<in>set (fresh_rename_idns n B insts G). nm \<notin> fst ` (fv B \<union> \<Union> (fv ` snd ` set insts) \<union> fst ` set insts) \<union> G"
le_diff: 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 "A \<noteq> B"   and "Tarski_neutral_dimensionless.Le Bet Cong A B C D" shows "C \<noteq> D"
\<I>2: shows "\<I>2 \<le> map_\<I> f2 g2 \<I>2'"
z_zero_not_in_S: fixes p :: "proj2" assumes "proj2_incident p z_zero" shows "p \<notin> S"
vec_to_iarray_least_non_zero_position_of_vector_from_index': fixes i :: "'b"   and j :: "'c"   and A :: "(('a, 'c) vec, 'b) vec" assumes "\<not> vector_all_zero_from_index (mod_type_class.to_nat i, vec_to_iarray (column j A))" shows "least_non_zero_position_of_vector_from_index (vec_to_iarray (column j A)) (mod_type_class.to_nat i) = mod_type_class.to_nat (LEAST n. A $ n $ j \<noteq> (0::'a) \<and> i \<le> n)"
tt_plus_lessE: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'f"   and q :: "'c \<Rightarrow>\<^sub>0 'f" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "p + q \<noteq> 0"   and "ord_term_strict (ordered_term.tt term_of_pair ord_term (p + q)) (ordered_term.tt term_of_pair ord_term p)" shows "ord_term_strict (ordered_term.tt term_of_pair ord_term q) (ordered_term.tt term_of_pair ord_term p)"
game_def: fixes \<A> :: "'grp \<Rightarrow> 'grp \<Rightarrow> 'grp set spmf" shows "game \<A> = sample_uniform (order \<G>) \<bind> (\<lambda>x. sample_uniform (order \<G>) \<bind> (\<lambda>y. \<A> (\<^bold>g [^] x) (\<^bold>g [^] y) \<bind> (\<lambda>zs. return_spmf (\<^bold>g [^] (x * y) \<in> zs))))"
same_prefix_nil: fixes xs :: "'a list"   and ys :: "'a list" shows "prefix (xs @ ys) xs = (ys = [])"
Rep_le_n: fixes x :: "'a" shows "Rep x \<le> n"
1: fixes x :: "real" assumes "0 < x"   and "x * 2 < pi" shows "(tan has_real_derivative 1 + (tan x)\<^sup>2) (at x)"
mult2_alt_s_ns_add: fixes X1 :: "'a multiset"   and Y1 :: "'a multiset"   and ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set"   and X2 :: "'a multiset"   and Y2 :: "'a multiset" assumes "(X1, Y1) \<in> mult2_alt_s ns s"   and "(X2, Y2) \<in> mult2_alt_ns ns s" shows "(X1 + X2, Y1 + Y2) \<in> mult2_alt_s ns s"
p: shows "bprv (imp (PP \<langle>\<phi>\<rangle>) \<phi>)"
Category_axioms_def: fixes C :: "('a, 'b, 'c) Category_scheme" shows "Category_axioms C \<equiv> ((\<forall>f. f \<in> mor\<^bsub>C\<^esub> \<longrightarrow> dom\<^bsub>C\<^esub> f \<in> obj\<^bsub>C\<^esub>) \<and> (\<forall>f. f \<in> mor\<^bsub>C\<^esub> \<longrightarrow> cod\<^bsub>C\<^esub> f \<in> obj\<^bsub>C\<^esub>) \<and> (\<forall>X. X \<in> obj\<^bsub>C\<^esub> \<longrightarrow> id\<^bsub>C\<^esub> X maps\<^bsub>C\<^esub> X to X)) \<and> ((\<forall>f. f \<in> mor\<^bsub>C\<^esub> \<longrightarrow> id\<^bsub>C\<^esub> (dom\<^bsub>C\<^esub> f) ;;\<^bsub>C\<^esub> f = f) \<and> (\<forall>f. f \<in> mor\<^bsub>C\<^esub> \<longrightarrow> f ;;\<^bsub>C\<^esub> id\<^bsub>C\<^esub> (cod\<^bsub>C\<^esub> f) = f)) \<and> (\<forall>f g h. f \<approx>>\<^bsub>C\<^esub> g \<longrightarrow> g \<approx>>\<^bsub>C\<^esub> h \<longrightarrow> f ;;\<^bsub>C\<^esub> g ;;\<^bsub>C\<^esub> h = f ;;\<^bsub>C\<^esub> (g ;;\<^bsub>C\<^esub> h)) \<and> (\<forall>f X Y g Z. f maps\<^bsub>C\<^esub> X to Y \<longrightarrow> g maps\<^bsub>C\<^esub> Y to Z \<longrightarrow> f ;;\<^bsub>C\<^esub> g maps\<^bsub>C\<^esub> X to Z)"
compseriesTr2: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set" assumes "compseries G n f" shows "f 0 = carrier G"
order_iff_strict: fixes b :: "'b multiset"   and a :: "'b multiset" shows "(b \<subseteq># a) = (b \<subset># a \<or> a = b)"
simps_4: fixes name :: "nat"   and incoming :: "nat set"   and new :: "'b ltlr set"   and old :: "'b ltlr set"   and "next" :: "'b ltlr set"   and more :: "'c" shows "old \<lparr>name = name, incoming = incoming, new = new, old = old, next = next, \<dots> = more\<rparr> = old"
add_comm: fixes x :: "'a"   and y :: "'a" shows "x + y = y + x"
hom_mult: fixes x :: "'a poly poly"   and y :: "'a poly poly" shows "poly_y_x (x * y) = poly_y_x x * poly_y_x y"
abs_eq: fixes x :: "'k \<Rightarrow> nat"   and xa :: "'k \<Rightarrow> bool" assumes "eq_onp (\<lambda>x. x \<in> multiset) x x" shows "filter_mset xa (Abs_multiset x) = Abs_multiset (\<lambda>x. if xa x then x x else 0)"
elims: fixes x :: "'b poly"   and y :: "nat \<Rightarrow> nat" assumes "degreen x = y" obtains c :: "'b poly" and n :: "nat" and p :: "'b poly" where "x = CN c n p" and "y = (\<lambda>m. if n = m then 1 + degreen p n else 0)"   | v :: "'b" where "x = (v__)\<^sub>p" and "y = (\<lambda>m. 0)"   | v :: "nat" where "x = Bound va__" and "y = (\<lambda>m. 0)"   | v :: "'b poly" and va :: "'b poly" where "x = Add vb__ vaa__" and "y = (\<lambda>m. 0)"   | v :: "'b poly" and va :: "'b poly" where "x = Sub vc__ vab__" and "y = (\<lambda>m. 0)"   | v :: "'b poly" and va :: "'b poly" where "x = Mul vd__ vac__" and "y = (\<lambda>m. 0)"   | v :: "'b poly" where "x = Neg ve__" and "y = (\<lambda>m. 0)"   | v :: "'b poly" and va :: "nat" where "x = Pw v va" and "y = (\<lambda>m. 0)"
finite_range: fixes y :: "'index \<times> 'trap" shows "y \<in> set_spmf I \<longrightarrow> finite (range (fst y))"
relpowp_0_I: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" shows "(P ^^ 0) x x"
bij_image_Compl_eq: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "bij f" shows "f ` (- A) = - f ` A"
circuit_subset_carrier: fixes X :: "'a set" assumes "circuit X" shows "X \<subseteq> carrier"
csucc_le: fixes \<mu> :: "V"   and \<kappa> :: "V" assumes "Card \<mu>"   and "\<kappa> \<in> elts \<mu>" shows "csucc \<kappa> \<le> \<mu>"
term_of_pair_pair: fixes v :: "'a" shows "fst (local.punit.pp_of_term v, local.punit.component_of_term v) = v"
wf\<^sub>s\<^sub>t_induct': fixes V :: "'c set"   and S :: "('d, 'c) strand_step list"   and P :: "('d, 'c) strand_step list \<Rightarrow> bool" assumes "wf\<^sub>s\<^sub>t V S"   and "P []"   and "\<And>t S. \<lbrakk>wf\<^sub>s\<^sub>t V S; P S\<rbrakk> \<Longrightarrow> P (S @ [send\<langle>t\<rangle>\<^sub>s\<^sub>t])"   and "\<And>t S. \<lbrakk>wf\<^sub>s\<^sub>t V S; P S; fv t \<subseteq> V \<union> wfvarsoccs\<^sub>s\<^sub>t S\<rbrakk> \<Longrightarrow> P (S @ [receive\<langle>t\<rangle>\<^sub>s\<^sub>t])"   and "\<And>t t' S. \<lbrakk>wf\<^sub>s\<^sub>t V S; P S; fv t' \<subseteq> V \<union> wfvarsoccs\<^sub>s\<^sub>t S\<rbrakk> \<Longrightarrow> P (S @ [\<langle>assign: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t])"   and "\<And>t t' S. \<lbrakk>wf\<^sub>s\<^sub>t V S; P S\<rbrakk> \<Longrightarrow> P (S @ [\<langle>check: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t])"   and "\<And>X F S. \<lbrakk>wf\<^sub>s\<^sub>t V S; P S\<rbrakk> \<Longrightarrow> P (S @ [\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t])" shows "P S"
tls1_to_tls2_def: fixes s2 :: "'s2"   and stlsss1 :: "('tl1 \<times> 's1, 's1 option) tllist" shows "tls1_to_tls2 s2 stlsss1 = unfold_tllist (\<lambda>(s2, stlsss1). is_TNil stlsss1) (\<lambda>(s2, stlsss1). map_option (\<lambda>s1'. SOME s2'. s2 -\<tau>2\<rightarrow>* s2' \<and> (\<forall>tl s2''. \<not> s2' -2-tl\<rightarrow> s2'') \<and> s1' \<approx> s2') (terminal stlsss1)) (\<lambda>(s2, stlsss1). let (tl2, s2'') = SOME (tl2, s2''). \<exists>s2'. s2 -\<tau>2\<rightarrow>* s2' \<and> s2' -2-tl2\<rightarrow> s2'' \<and> \<not> \<tau>move2 s2' tl2 s2'' \<and> snd (thd stlsss1) \<approx> s2'' \<and> fst (thd stlsss1) \<sim> tl2 in (tl2, s2'')) (\<lambda>(s2, stlsss1). let (tl2, s2'') = SOME (tl2, s2''). \<exists>s2'. s2 -\<tau>2\<rightarrow>* s2' \<and> s2' -2-tl2\<rightarrow> s2'' \<and> \<not> \<tau>move2 s2' tl2 s2'' \<and> snd (thd stlsss1) \<approx> s2'' \<and> fst (thd stlsss1) \<sim> tl2 in (s2'', ttl stlsss1)) (s2, stlsss1)"
deriv2: fixes y :: "real" shows "(Q (n - Suc k) has_real_derivative Q (n - k) y) (at y within {0..1})"
fst_app_def: fixes t :: "lam" shows "fst_app t = the (fst_app_aux t)"
valid_path_aux_valid_path: fixes cs :: "('b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node) list"   and as :: "('b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node) list" assumes "CFGExit.valid_path_aux cs as" shows "CFGExit.valid_path as"
in_H_imp_eq: fixes g :: "'a" assumes "g \<in> H" shows "M1 #> g = M1"
New: fixes s :: "st"   and a :: "loc"   and x :: "abopt"   and oi' :: "obj_tag"   and oi :: "obj_tag"   and G :: "prog" assumes "new_Addr (heap s) = Some a"   and "(x, oi') = (if atleast_free (heap s) two then (None, oi) else (Some (Xcpt (Loc a)), CInst (SXcpt OutOfMemory)))" shows "G\<turnstile>Norm s \<midarrow>halloc oi\<succ>a\<rightarrow> (x, init_obj G oi' (Inl a) s)"
cr_prod1_simps: fixes c' :: "'d"   and A :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'e"   and b :: "'f"   and c :: "'d" shows "cr_prod1 c' A a (b, c) = (A a b \<and> c' = c)"
disc_23: fixes x31 :: "nat"   and x32 :: "'d"   and x33 :: "'d"   and x34 :: "'e"   and x35 :: "'e"   and x36 :: "'f" shows "\<not> isRecvMarker (Recv x31 x32 x33 x34 x35 x36)"
add_strict_right_mono: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset" assumes "a \<subset># b" shows "a + c \<subset># b + c"
eq_divide_eq_numeral1_1: fixes a :: "'c"   and b :: "'c"   and w :: "num" shows "(a = b / numeral w) = (if numeral w \<noteq> (0::'c) then a * numeral w = b else a = (0::'c))"
sum_simp_setup_6: shows "\<forall>v\<in>V. sum (augment f') (outgoing v) = (\<Sum>u\<in>V. augment f' (v, u))"
fseq: shows "(\<Prod>(a, i)\<in>set fs. a ^ Suc i) = (\<Prod>(a, i)\<in>set fs'. normalize a ^ Suc i)"
minimalize_offending_overapprox_subset: fixes ff :: "('v \<times> 'v) list"   and keeps :: "('v \<times> 'v) list"   and G :: "'v graph"   and nP :: "'v \<Rightarrow> 'a" shows "set (minimalize_offending_overapprox ff keeps G nP) \<subseteq> set ff \<union> set keeps"
pred_of_enex_def: fixes enex :: "('b \<Rightarrow> 'e set) \<times> ('e \<Rightarrow> 'b \<Rightarrow> 'd)" shows "pred_of_enex enex \<equiv> \<lambda>s s'. let (en, ex) = enex in \<exists>a\<in>en s. s' = ex a s"
\<gamma>_lt_\<delta>: shows "\<gamma> < \<delta>"
that_9: fixes \<Gamma>' :: "char list \<Rightarrow> secLevel option"   and c :: "com" assumes "x = \<Gamma>'"   and "xa = Low"   and "xb = c"   and "\<Gamma>',High \<turnstile> c" shows "thesis"
IH: fixes y :: "nat"   and S :: "'a set" assumes "y < n"   and "dim S = y"   and "compact S"   and "convex S"   and "(0::'a) \<in> S" shows "(0::'a) \<in> convex hull {x. x extreme_point_of S}"
solve_consistent_rref_iarrays_def: fixes A :: "'e iarray iarray"   and b :: "'d iarray" shows "solve_consistent_rref_iarrays A b = IArray.of_fun (\<lambda>j. if IArray.exists (\<lambda>i. A !! i !! j = (1::'e) \<and> j = least_non_zero_position_of_vector (row_iarray i A)) (IArray [0..<nrows_iarray A]) then b !! least_non_zero_position_of_vector (column_iarray j A) else (0::'d)) (ncols_iarray A)"
f1: shows "(Matrix.mat (2 ^ n) (2 ^ n) (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> (- 1) ^ i \<cdot>\<^bsub>n\<^esub>  j / sqrt 2 ^ n)) \<Otimes> Quantum.Id 1) $$ (i, j) = Matrix.mat (2 ^ n) (2 ^ n) (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> (- 1) ^ i \<cdot>\<^bsub>n\<^esub>  j / sqrt 2 ^ n)) $$ (i div dim_row (Quantum.Id 1), j div dim_col (Quantum.Id 1)) * Quantum.Id 1 $$ (i mod dim_row (Quantum.Id 1), j mod dim_col (Quantum.Id 1))"
distinct_append_butlast1: fixes xs :: "('a \<times> 'b) list"   and ys :: "('a \<times> 'c) list" assumes "distinct (map fst xs @ map fst ys)" shows "distinct (map fst (butlast xs) @ map fst ys)"
safe_def: fixes s :: "event list"   and r :: "room" shows "Trace.safe s r = (\<exists>s\<^sub>1 s\<^sub>2 s\<^sub>3 g c. s = s\<^sub>3 @ [Enter g r c] @ s\<^sub>2 @ [Check_in g r c] @ s\<^sub>1 \<and> no_Check_in (s\<^sub>3 @ s\<^sub>2) r \<and> Trace.isin (s\<^sub>2 @ [Check_in g r c] @ s\<^sub>1) r = {})"
w_prop: shows "\<forall>y<w. - b * y + c < y\<^sup>2"
not_numeral_le_neg_numeral: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and m :: "num"   and n :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "\<not> less_eq (numeral.numeral one plus m) (uminus (numeral.numeral one plus n))"
assms_2: shows "\<S>\<bullet>\<C> \<turnstile> vs : [] _> ts @ [t]"
order_less_le_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x < y"   and "y \<le> z" shows "x < z"
select_convs_3: fixes getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and getConflictFlag :: "bool"   and getC :: "Literal list"   and more :: "'a" shows "getConflictFlag \<lparr>getF = getF, getM = getM, getConflictFlag = getConflictFlag, getC = getC, \<dots> = more\<rparr> = getConflictFlag"
Gr_univalent: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "univalent (BNF_Def.Gr A f)"
ring_id_closed: shows "ring_id \<in> carrier (function_ring (carrier R) R)"
degree_minus: fixes p :: "'b poly" shows "degree (- p) = degree p"
fac_final_def: shows "fac_final \<equiv> \<Lambda> n. unbox\<cdot>n >>= fac_work_final >>= box"
mult_minus_left: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.ring plus zero minus uminus times" shows "times (uminus a) b = uminus (times a b)"
S_TopE: fixes \<Gamma> :: "binding list"   and T :: "ty" assumes "\<Gamma>\<turnstile>Top<:T" obtains "T = Top" and "\<turnstile> \<Gamma> ok" and "Top closed_in \<Gamma>"
deriv_Q: shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
Gromov_extension_isometry_inv_1: fixes f :: "'b \<Rightarrow> 'c" assumes "isometry f" shows "inv (Gromov_extension f) = Gromov_extension (inv f)"
length_snoc: fixes xs :: "'a list"   and x :: "'a" shows "length (xs @ [x]) = length xs + 1"
f_join_f_shrink_iT_Plus_iT_Div_mod_subst: fixes k :: "nat"   and I :: "nat set"   and A :: "nat set"   and B :: "nat set"   and xs :: "'b message_af list" assumes "0 < k"   and "\<forall>x\<in>I. x mod k = 0"   and "A = I \<oplus> k - 1"   and "B = I \<oslash> k" shows "xs \<longmapsto> k \<Join>\<^sub> A = xs \<div> k \<Join>\<^sub> B"
strongBisimWeakCong: fixes P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "P \<sim>\<^sub>s Q" shows "P \<doteq>\<^sub>c Q"
antisym_conv1: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.order less_eq less"   and "\<not> less x y" shows "less_eq x y = (x = y)"
AGM_FullContraction_def: fixes Cn :: "'b set \<Rightarrow> 'b set"   and contraction :: "'b set \<Rightarrow> 'b \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "AGM_FullContraction Cn contraction true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL \<equiv> AGM_Contraction Cn contraction \<and> Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL \<and> AGM_FullContraction_axioms Cn contraction conj_PL"
as_def: shows "as = (\<Prod>j = 1..s - 1. real_of_int (a j))"
n_preserve2: fixes x :: "'a"   and y :: "'a" shows "n x \<cdot> n y \<cdot> t x = o"
intros_1: fixes F :: "\<Pi>\<^sub>1" shows "SimpleExOrEnc (exe1 F)"
split_pi_off: shows "x + real (Suc n) * (2 * pi) = x + real n * (2 * pi) + 2 * pi"
typeof_conf: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and v :: "val"   and T :: "ty"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list" assumes "typeof\<^bsub>h\<^esub> v = \<lfloor>T\<rfloor>" shows "P,h \<turnstile> v :\<le> T"
map_def: fixes f :: "'a \<times> 'a" shows "\<Pi>o\<pi>_\<delta>o\<Pi>.map f = (if CC.arr f then (local.Prod \<circ> \<pi>.map) (CC.cod f) \<cdot> (dup \<circ> local.Prod) f else null)"
qtable2: shows "qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) (ne + length_muaux args aux) \<psi>) rel2"
a: shows "(SOME x. x = False \<or> x = True \<and> A) = False \<or> (SOME x. x = False \<or> x = True \<and> A) = True \<and> A"
homotopic_on_empty: fixes X :: "'b topology"   and P :: "('b \<Rightarrow> 'c) \<Rightarrow> bool"   and X' :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "topspace X = {}" shows "homotopic_with P X X' f g = (P f \<and> P g)"
hta_bwd_reduce_def: fixes H :: "('a, 'b, 'c) hashedTa_scheme" shows "hta_bwd_reduce H \<equiv> let rqrm = build_rqrm (hta_\<delta> H) in hta_reduce H (fst (while brc_cond (brc_step rqrm) (brc_initial (hta_\<delta> H))))"
lubIa: fixes S :: "'a set"   and L :: "'a" assumes "S \<subseteq> A"   and "isLub S cl L" shows "L = lub S cl"
pair'_def: fixes pair_fun :: "'a"   and d :: "('a, 'b) Term.term \<times> ('a, 'b) Term.term" shows "pair' pair_fun d \<equiv> case d of (t, t') \<Rightarrow> Fun pair_fun [t, t']"
crypts_fst: fixes KS :: "key list"   and X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "foldr Crypt KS \<lbrace>X, Y\<rbrace> \<in> crypts H" shows "foldr Crypt KS X \<in> crypts H"
finer_partJoin_L: fixes I :: "'a set"   and P :: "'a set set"   and Q :: "'a set set" assumes "part I P"   and "part I Q" shows "finer P (partJoin P Q)"
uminus_in_iff: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" shows "((\<lambda>x. - f x) \<in> o[F](g)) = (f \<in> o[F](g))"
simps_2: fixes V :: "'c \<Rightarrow> 'd list"   and k :: "'c"   and ks :: "'c list" shows "mapping_generator V (k # ks) = {Mapping.update k v m |v m. v \<in> set (V k) \<and> m \<in> mapping_generator V ks}"
gfp_exp_def: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "gfp_exp t = Sup_exp {P. unitary P \<and> P \<tturnstile> t P}"
sympE: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and b :: "'c"   and a :: "'c" assumes "symp r"   and "r b a" obtains "r a b"