hcju/isabelleps huggingface dataset:
safe_formula_induct: fixes \<phi> :: "'a MFOTL.formula"   and P :: "'a MFOTL.formula \<Rightarrow> bool" assumes "safe_formula \<phi>"   and "\<And>t1 t2. trm.is_Const t1 \<Longrightarrow> P (formula.Eq t1 t2)"   and "\<And>t1 t2. trm.is_Const t2 \<Longrightarrow> P (formula.Eq t1 t2)"   and "\<And>x y. P (formula.Neg (formula.Eq (trm.Const x) (trm.Const y)))"   and "\<And>x y. x = y \<Longrightarrow> P (formula.Neg (formula.Eq (trm.Var x) (trm.Var y)))"   and "\<And>e ts. P (formula.Pred e ts)"   and "\<And>\<phi> \<psi>. \<lbrakk>\<not> (safe_formula (formula.Neg \<psi>) \<and> fv \<psi> \<subseteq> fv \<phi>); P \<phi>; P \<psi>\<rbrakk> \<Longrightarrow> P (MFOTL.And \<phi> \<psi>)"   and "\<And>\<phi> \<psi>. \<lbrakk>safe_formula \<psi>; fv \<psi> \<subseteq> fv \<phi>; P \<phi>; P \<psi>\<rbrakk> \<Longrightarrow> P (MFOTL.And_Not \<phi> \<psi>)"   and "\<And>\<phi> \<psi>. \<lbrakk>fv \<psi> = fv \<phi>; P \<phi>; P \<psi>\<rbrakk> \<Longrightarrow> P (formula.Or \<phi> \<psi>)"   and "\<And>\<phi>. P \<phi> \<Longrightarrow> P (formula.Exists \<phi>)"   and "\<And>I \<phi>. P \<phi> \<Longrightarrow> P (formula.Prev I \<phi>)"   and "\<And>I \<phi>. P \<phi> \<Longrightarrow> P (formula.Next I \<phi>)"   and "\<And>\<phi> I \<psi>. \<lbrakk>fv \<phi> \<subseteq> fv \<psi>; safe_formula \<phi>; P \<phi>; P \<psi>\<rbrakk> \<Longrightarrow> P (formula.Since \<phi> I \<psi>)"   and "\<And>\<phi> I \<psi>. \<lbrakk>fv (formula.Neg \<phi>) \<subseteq> fv \<psi>; \<not> safe_formula (formula.Neg \<phi>); P \<phi>; P \<psi>\<rbrakk> \<Longrightarrow> P (formula.Since (formula.Neg \<phi>) I \<psi>)"   and "\<And>\<phi> I \<psi>. \<lbrakk>fv \<phi> \<subseteq> fv \<psi>; safe_formula \<phi>; P \<phi>; P \<psi>\<rbrakk> \<Longrightarrow> P (formula.Until \<phi> I \<psi>)"   and "\<And>\<phi> I \<psi>. \<lbrakk>fv (formula.Neg \<phi>) \<subseteq> fv \<psi>; \<not> safe_formula (formula.Neg \<phi>); P \<phi>; P \<psi>\<rbrakk> \<Longrightarrow> P (formula.Until (formula.Neg \<phi>) I \<psi>)" shows "P \<phi>"
eps: fixes i :: "'a"   and j :: "'b" assumes "i \<in> I"   and "j \<in> J"   and "w i \<noteq> 0"   and "v j \<noteq> 0" shows "dist (f i) (g j) \<le> d i + e j"
lookupDef_looksup': fixes v' :: "'var"   and n' :: "'node"   and def :: "Def"   and g :: "'g"   and n :: "'node"   and v :: "'var" assumes "(v', n', def) = lookupDef g n v"   and "n \<in> set (\<alpha>n g)" shows "v' = v"
le_the_leftI: fixes i :: "nat"   and j :: "nat" assumes "i \<le> j"   and "xs ! i < xs ! j" shows "i \<le> local.the_left j"
indep_region_coregion: fixes S :: "<'d, 'e> \<Longleftrightarrow> 'f" assumes "sym_lens S" shows "\<V>\<^bsub>S\<^esub> \<bowtie> \<C>\<^bsub>S\<^esub>"
ph: fixes j :: "nat"   and v :: "nat" shows "eval ph [j, v, i, e] = \<phi> i j"
cSum1: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and C :: "'a"   and Q :: "pi" assumes "P \<longmapsto> a<\<nu>x> \<prec> P'"   and "\<And>C. F C P a x P'" shows "F C (P \<oplus> Q) a x P'"
SN_imp_WN: fixes A :: "('c \<times> 'c) set" assumes "SN A" shows "WN A"
relcomp_subset_Sigma: fixes r :: "('d \<times> 'e) set"   and A :: "'d set"   and B :: "'e set"   and s :: "('e \<times> 'f) set"   and C :: "'f set" assumes "r \<subseteq> A \<times> B"   and "s \<subseteq> B \<times> C" shows "r O s \<subseteq> A \<times> C"
games_lfp_represent: fixes x :: "ZF" assumes "x \<in> games_lfp" shows "\<exists>l r. x = Opair l r"
axioms_3: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<BB>" shows "tiny_digraph \<alpha> \<BB>"
unify_complete: fixes E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list"   and bs :: "('c \<times> ('d, 'c) Term.term) list" assumes "unify E bs = None" shows "unifiers (set E) = {}"
simps_10: fixes v :: "v" shows "do_opapp [v] = None"
below_sctn_nres: fixes X :: "'c set"   and sctn :: "'c sctn" shows "Sup_inner X (normal sctn) \<bind> (\<lambda>si. RETURN (si \<le> pstn sctn)) \<le> below_sctn X sctn"
assms_3: shows "u \<in> nodes g\<^sub>1"
ipurge_ref_aux_foldr_subset: fixes I :: "('e \<times> 'e) set"   and D :: "'f \<Rightarrow> 'e"   and U :: "'e set"   and ys :: "'f list"   and V :: "'e set"   and P :: "'f \<Rightarrow> bool"   and xs :: "'f list"   and X :: "'f set" assumes "sinks_aux I D U ys \<subseteq> V" shows "ipurge_ref_aux I D U ys (ipurge_ref_aux_foldr I D P V xs X) = ipurge_ref_aux_foldr I D P V xs X"
term_cases: fixes t :: "'a" obtains name :: "name" where "t = free name__"   | name :: "name" where "t = const name"   | u\<^sub>1 :: "'a" and u\<^sub>2 :: "'a" where "t = app u\<^sub>1 u\<^sub>2"   | "unfree t = None" and "unapp t = None" and "unconst t = None"
induct: fixes P :: "[:'a:] \<Rightarrow> bool"   and x :: "[:'a:]" assumes "P \<bottom>"   and "P [::]"   and "\<And>a slist. \<lbrakk>a \<noteq> \<bottom>; slist \<noteq> \<bottom>; P slist\<rbrakk> \<Longrightarrow> P (a :# slist)" shows "P x"
get_option_def: fixes get :: "('i \<Rightarrow> 'h) \<Rightarrow> 'h"   and f :: "'i \<Rightarrow> ('g, 'h) optionT" shows "get_option get f = OptionT (get (\<lambda>s. run_option (f s)))"
connexI: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> less_eq x y \<or> less_eq y x" shows "connex A less_eq"
c3_1: fixes x :: "real"   and i :: "nat" assumes "x\<^sub>1 \<le> x"   and "i < k" shows "c3 * g x \<le> g_approx i x"
correct_routing: fixes rtbl :: "'i routing_rule list" assumes "rtblo = Some rtbl" shows "correct_routing rtbl"
SumGoal: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" assumes "P \<in> summands Q"   and "uhnf (Q \<oplus> R)"   and "\<And>P. P \<in> summands Q \<Longrightarrow> \<exists>Q'. P \<oplus> Q' \<equiv>\<^sub>e Q \<and> summands Q' = summands Q - {P' |P'. P' \<in> summands Q \<and> P' \<equiv>\<^sub>e P} \<and> uhnf Q'"   and "\<And>P. P \<in> summands R \<Longrightarrow> \<exists>R'. P \<oplus> R' \<equiv>\<^sub>e R \<and> summands R' = summands R - {P' |P'. P' \<in> summands R \<and> P' \<equiv>\<^sub>e P} \<and> uhnf R'" shows "\<exists>Q'. P \<oplus> Q' \<equiv>\<^sub>e Q \<oplus> R \<and> summands Q' = summands (Q \<oplus> R) - {P' |P'. P' \<in> summands (Q \<oplus> R) \<and> P' \<equiv>\<^sub>e P} \<and> uhnf Q'"
closedin_Int_closure_of: fixes X :: "'b topology"   and S :: "'b set"   and T :: "'b set" shows "closedin (subtopology X S) T = (S \<inter> X closure_of T = T)"
subcls1_irrefl: fixes x :: "qtname"   and y :: "qtname"   and G :: "prog" assumes "G\<turnstile>x\<prec>\<^sub>C1y"   and "ws_prog G" shows "x \<noteq> y"
INF_constant: fixes c :: "'al"   and U\<^sub>2 :: "'c set"   and f :: "'c \<Rightarrow> 'al"   and A :: "'c set" assumes "c \<in> U"   and "\<forall>x\<in>U\<^sub>2. f x = c"   and "A \<subseteq> U\<^sub>2" shows "\<Sqinter>\<^sub>o\<^sub>w (f ` A) = (if A = {} then \<top>\<^sub>o\<^sub>w else c)"
prv_imp_cnj4: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi>1 :: "'fmla"   and \<psi>2 :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi>1 \<in> fmla"   and "\<psi>2 \<in> fmla"   and "prv (imp \<phi> (imp \<chi> \<psi>1))"   and "prv (imp \<phi> (imp \<chi> \<psi>2))" shows "prv (imp \<phi> (imp \<chi> (cnj \<psi>1 \<psi>2)))"
dual_complemented_lattice: shows "class.complemented_lattice (\<lambda>x y. x \<squnion> - y) uminus (\<squnion>) (\<lambda>x y. y \<le> x) (\<lambda>x y. y < x) (\<sqinter>) \<top> \<bottom>"
psubst_fls: fixes txs :: "('trm \<times> 'var) list" assumes "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm" shows "psubst fls txs = fls"
ax3: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" assumes "|~ F \<and> Unchanged v \<longrightarrow> \<circle>F" shows "\<turnstile> \<box>[F \<longrightarrow> \<circle>F]_v \<longrightarrow> F \<longrightarrow> \<box>F"
prems: shows "\<forall>a\<in>ran \<Gamma> \<union> {rs}. wf_chain \<Gamma> a"
atom_ordering_total: fixes atom_ordering :: "('a \<times> 'a) set" assumes "propositional_atoms atom_ordering" shows "\<forall>x y. x \<noteq> y \<longrightarrow> (x, y) \<in> atom_ordering \<or> (y, x) \<in> atom_ordering"
wI_2: shows "wf_interp_for_formula (w, I) (FOr \<phi> \<psi>)"
AbstrLevels_S4opt_A31: fixes i :: "AbstrLevelsID" assumes "sS4opt \<in> AbstrLevel i" shows "sA31 \<notin> AbstrLevel i"
no_before_inI: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list" assumes "x < y in xs" shows "(\<not> y < x in xs) = True"
awalk_verts_append: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and q :: "('a \<times> 'a) list"   and v :: "'a" assumes "awalk u (p @ q) v" shows "pawalk_verts u (p @ q) = pawalk_verts u p @ tl (pawalk_verts (last (pawalk_verts u p)) q)"
3: shows "s \<preceq> s1 \<sqinter> s2"
distribute_rely_conjunction: fixes i :: "'a::refinement_lattice"   and c :: "'a::refinement_lattice"   and d :: "'a::refinement_lattice" assumes "chaos \<sqsubseteq> (i::'a::refinement_lattice)" shows "((c::'a::refinement_lattice) \<iinter> (d::'a::refinement_lattice)) // (i::'a::refinement_lattice) \<sqsubseteq> c // i \<iinter> d // i"
prio_def: fixes SkewBinomialTree :: "('d, 'c) SkewBinomialTree" shows "prio SkewBinomialTree = (case SkewBinomialTree of Node x1 x2 x3 x4 \<Rightarrow> x2)"
between_antisym: fixes b :: "'a"   and c :: "'a"   and a :: "'a" assumes "between (b, c) a"   and "between (a, c) b" shows "a = b"
add_uminus_conv_diff: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.group_add minus plus zero uminus" shows "plus a (uminus b) = minus a b"
less_le_not_le: fixes y :: "'d"   and x :: "'d" shows "(y < x) = (y \<le> x \<and> \<not> x \<le> y)"
cut_less_Int_left: fixes A :: "'a set"   and B :: "'a set"   and t :: "'a" shows "(A \<inter> B) \<down>< t = A \<down>< t \<inter> B"
g_not_f: fixes x :: "'b"   and y :: "nat"   and z :: "nat" assumes "(x, y) \<in> (let V_new = sorted_list_of_set (vertices (snd R) - vertices (fst R)) in set (zip V_new [n..<n + length V_new]))" shows "(x, z) \<notin> f"
add: fixes b1 :: "'a"   and b2 :: "'a"   and y :: "'b" shows "h (b1 + b2) y = h b1 y + h b2 y"
has_integral_integral_lborel: fixes f :: "'c \<Rightarrow> 'd" assumes "integrable lborel f" shows "(f has_integral integral\<^sup>L lborel f) UNIV"
reachable_trace: fixes q :: "'a"   and A :: "('b, 'a) nba"   and p :: "'a"   and r :: "('b \<times> 'a) stream" assumes "q \<in> reachable A p"   and "run A r q" shows "sset (trace r q) \<subseteq> reachable A p"
stutter_extend_edges_def: fixes V :: "'b set"   and E :: "('b \<times> 'b) set" shows "stutter_extend_edges V E \<equiv> E \<union> {(v, v) |v. v \<in> V \<and> v \<notin> Domain E}"
connected_component_eq_self: fixes S :: "'a set"   and x :: "'a" assumes "connected S"   and "x \<in> S" shows "connected_component_set S x = S"
simE_1: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and a :: "name"   and x :: "name"   and Q' :: "pi" assumes "P \<leadsto><Rel> Q"   and "Q \<longmapsto> a<\<nu>x> \<prec> Q'"   and "x \<sharp> P" shows "\<exists>P'. P \<Longrightarrow>\<^sub>l a<\<nu>x> \<prec> P' \<and> (P', Q') \<in> Rel"
hyps_3: shows "((sh @ r, ch), w, s' @ r, c') \<in> trcl (trss fg)"
coerce_def: shows "COERCE('d, 'e) \<equiv> PRJ('e) oo EMB('d)"
IH: assumes "generalized_sfw (generalized_fw_join as f2) p = Some (u, d1, d2)" shows "\<exists>r1 r2. generalized_sfw as p = Some (r1, d1) \<and> generalized_sfw f2 p = Some (r2, d2) \<and> Some u = simple_match_and r1 r2"
new_element_get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_element_ptr :: "(_) element_ptr"   and ptr :: "(_) object_ptr"   and getter :: "(_) RObject_ext \<Rightarrow> 'j" assumes "h \<turnstile> new_element \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr"   and "ptr \<noteq> cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"
hensel_fpxs1_def: shows "hensel_fpxs1 = Abs_fps (fst \<circ> hensel_fpxs_aux)"
fls_deriv_fls_integral: fixes a :: "'b fls" assumes "fls_residue a = (0::'b)" shows "fls_deriv (fls_integral a) = a"
ln_2_le: shows "ln 2 \<le> 355 / 512"
remove: fixes x :: "'a"   and A :: "'a multiset" assumes "x \<in># A" shows "prod_mset A = x * prod_mset (A - {#x#})"
hyps_1: fixes \<AA> :: "'i" assumes "m = Length \<AA>" shows "\<exists>xs \<BB>. \<AA> = fold CONS xs \<BB> \<and> Length \<BB> = 0 \<and> Length \<AA> = length xs \<and> #\<^sub>V \<BB> = #\<^sub>V \<AA> \<and> (\<forall>x\<in>set xs. size x = #\<^sub>V \<AA>)"
eval_elim_cases_18: fixes G :: "prog"   and s :: "st"   and D :: "qtname"   and c :: "stmt"   and x :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In1l (Body D c)\<succ>\<rightarrow> (x, s')" obtains s1 :: "state" and s2 :: "state" where "x = In1 (the (locals (snd s2) Result))" and "s' = abupd (absorb Ret) (if \<exists>l. fst s2 = Some (Jump (Break l)) \<or> fst s2 = Some (Jump (Cont l)) then abupd (\<lambda>x. Some (Error CrossMethodJump)) s2 else s2)" and "G\<turnstile>Norm s \<midarrow>Init D\<rightarrow> s1" and "G\<turnstile>s1 \<midarrow>c\<rightarrow> s2"
tt_min: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" assumes "lookup p v \<noteq> (0::'c)" shows "tt p \<preceq>\<^sub>t v"
set_trie_insert: fixes xs :: "bool list"   and t :: "trie" shows "set_trie (insert xs t) = set_trie t \<union> {xs}"
pw_eq_iff: fixes S :: "'c nres"   and S' :: "'c nres" shows "(S = S') = (nofail S = nofail S' \<and> (\<forall>x. inres S x = inres S' x))"
flrestriction_fid_on: fixes A :: "V"   and B :: "V" shows "fid_on A \<restriction>\<^sup>l\<^sub>\<bullet> B = fid_on (A \<inter>\<^sub>\<circ> B)"
Re_express_cnj: fixes z :: "complex" shows "cor (Re z) = (z + cnj z) / 2"
square_free_mod_imp_square_free: fixes p :: "int"   and f :: "int poly" assumes "prime p"   and "poly_mod.square_free_m p f"   and "comm_monoid_mult_class.coprime (lead_coeff f) p" shows "square_free f"
f: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "('a \<times> 'a) \<times> 'a \<times> 'a \<Rightarrow> ennreal"   and S :: "('a \<times> 'a) set" assumes "web_flow (web_of_network \<Delta>) f"   and "orthogonal_current (web_of_network \<Delta>) f S"   and "separating (web_of_network \<Delta>) S"   and "\<And>e. e \<in> \<^bold>E\<^bsub>\<Delta>\<^esub> \<Longrightarrow> 0 < capacity \<Delta> e" shows "current (web_of_network \<Delta>) f"
span_eq_iff: fixes s :: "(('a \<Rightarrow> real) \<Rightarrow> real) set"   and p :: "'a" assumes "s \<subseteq> sub.tangent_space p" shows "(span s = s) = subspace s"
per_all: shows "\<forall>m\<in>{1..n}. periodic_arithmetic (\<lambda>k. \<chi> m * unity_root n (int (m * k))) n"
uminus_uexpr_def: fixes u :: "('e, 'f) uexpr" shows "- u = uop uminus u"
linear_eq: fixes var :: "nat"   and L :: "real list"   and C :: "real"   and b :: "real mpoly"   and c :: "real mpoly"   and B :: "real"   and p :: "real mpoly"   and v :: "real" assumes "var < length L"   and "C \<noteq> 0"   and "var \<notin> vars b"   and "var \<notin> vars c"   and "insertion (nth_default 0 (L[var := B / C])) b = B"   and "insertion (nth_default 0 (L[var := B / C])) c = C" shows "aEval (Eq p) (L[var := B / C]) = aEval (linear_substitution var b c (Eq p)) (L[var := v])"
inject: fixes x2 :: "n"   and y2 :: "n" shows "(S x2 = S y2) = (x2 = y2)"
LNil_Lazy_llist: shows "LNil = Lazy_llist Map.empty"
finite_ord_FO_resolution_inferences_between: fixes CC :: "'a literal multiset set"   and S :: "'a literal multiset \<Rightarrow> 'a literal multiset"   and C :: "'a literal multiset" assumes "finite CC" shows "finite (inference_system.inferences_between (ord_FO_\<Gamma> S) CC C)"
f_not_zero: shows "monom (UP R) \<one>\<^bsub>R\<^esub> n \<ominus>\<^bsub>UP R\<^esub> monom (UP R) (int (c mod p)) 0 \<noteq> \<zero>\<^bsub>UP R\<^esub>"
same_i_append_eq: fixes xs :: "'b list"   and f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" shows "(xs \<frown> f = xs \<frown> g) = (f = g)"
norm_minus_commute: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and a :: "'e"   and b :: "'e" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm (minus a b) = norm (minus b a)"
asBinp_def: fixes qbinp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qAbs option" shows "asBinp qbinp \<equiv> lift asAbs qbinp"
restricted_subformulas_W_\<mu>: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "subformulas\<^sub>\<mu> \<phi> \<subseteq> restricted_subformulas (\<phi> W\<^sub>n \<psi>)"
mod_eq_divisor_minus_Suc_0_conv: fixes k :: "nat"   and x :: "nat" assumes "Suc 0 < k" shows "(x mod k = k - Suc 0) = (Suc x mod k = 0)"
dvd_imp_mult_div_cancel_left: fixes a :: "'a"   and b :: "'a" assumes "a dvd b" shows "a * (b div a) = b"
Ord_not_le: fixes x :: "V"   and y :: "V" assumes "Ord x"   and "Ord y" shows "(\<not> x \<le> y) = (y < x)"
sum_list_distinct_conv_sum_set: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and xs :: "'b list"   and f :: "'b \<Rightarrow> 'c" assumes "class.monoid_add plus zero"   and "distinct xs" shows "sum_list (map f xs) = sum f (list.set xs)"
sylow_central_axioms: shows "sylow_central G p a m M1 M"
gcd_proj2_if_dvd: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times b a" shows "gcd a b = normalize b"
W: shows "{0, 1} \<times> S \<union> {0..1} \<times> T \<subseteq> W"
zero_reorient: fixes x :: "'b" shows "((0::'b) = x) = (x = (0::'b))"
inter_insert_fset: fixes x :: "'b"   and S :: "'b fset"   and T :: "'b fset" shows "insert_fset x S |\<inter>| T = (if x |\<in>| T then insert_fset x (S |\<inter>| T) else S |\<inter>| T)"
igSubstIGAbs_fromMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "gSubstGAbs MOD" shows "igSubstIGAbs (fromMOD MOD)"
esssup_eq: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure" assumes "f \<in> borel_measurable M" shows "esssup M f = Inf {z. emeasure M {x \<in> space M. z < f x} = 0}"
minus_cancel_vec: fixes v :: "'a vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "v - v = 0\<^sub>v n"
take_all_iff: fixes n :: "nat"   and xs :: "'b list" shows "(take n xs = xs) = (\<^bold>|xs\<^bold>| \<le> n)"
degree_eq_Suc_max: fixes x :: "'b pdevs" shows "degree x = (if \<forall>i. pdevs_apply x i = (0::'b) then 0 else Suc (Max {i. pdevs_apply x i \<noteq> (0::'b)}))"
Zfun_add: fixes f :: "'d \<Rightarrow> 'e"   and F :: "'d filter"   and g :: "'d \<Rightarrow> 'e" assumes "Zfun f F"   and "Zfun g F" shows "Zfun (\<lambda>x. f x + g x) F"
Abs_lifted_distrib_lattice_top_cases: fixes x :: "('c, 'd) lifted_distrib_lattice_top" obtains y :: "('c, 'd) phi \<Rightarrow> 'd" where "x = Abs_lifted_distrib_lattice_top y" and "y \<in> {xf. True}"
big_refl: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" shows "f \<in> \<Omega>[F](f)"
sum_in_spanI: fixes q :: "'c \<Rightarrow> 'c"   and B :: "'c set" shows "(\<Sum>b\<in>B. q b * b) \<in> ideal B"
circle_bot_edge_def: shows "circle_bot_edge = (1, \<lambda>t. (x_coord t * d, - d * circle_y (x_coord t)))"
l2_11: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p" assumes "Bet A B C"   and "Bet A' B' C'"   and "Cong A B A' B'"   and "Cong B C B' C'" shows "Cong A C A' C'"
m_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier C"   and "y \<in> carrier C"   and "z \<in> carrier C" shows "x \<otimes>\<^bsub>C\<^esub> y \<otimes>\<^bsub>C\<^esub> z = x \<otimes>\<^bsub>C\<^esub> (y \<otimes>\<^bsub>C\<^esub> z)"
fold_graph_fold: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'a set"   and z :: "'b" assumes "comp_fun_commute f"   and "finite A" shows "fold_graph f z A (Finite_Set.fold f z A)"
as_needed_asses_submap_exec_ix: fixes f :: "('d, 'e) fmap"   and g :: "('d, 'e) fmap" shows "(f \<subseteq>\<^sub>f g) = (\<forall>v\<in>fmdom' f. fmlookup f v = fmlookup g v)"
induct: fixes P1 :: "trm \<Rightarrow> bool"   and P3 :: "(char list \<times> trm) list \<Rightarrow> bool"   and P2 :: "char list \<times> trm \<Rightarrow> bool"   and char_list_trm_prod_list :: "(char list \<times> trm) list" assumes "\<And>x. P1 (Var x)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (\<lambda>:x1. x2)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (\<lambda><:x1. x2)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (x1 \<bullet> x2)"   and "\<And>x1 x2. P1 x1 \<Longrightarrow> P1 (x1 \<bullet>\<^sub>\<tau> x2)"   and "\<And>x. P3 x \<Longrightarrow> P1 (Rcd x)"   and "\<And>x1 x2. P1 x1 \<Longrightarrow> P1 (x1..x2)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x2; P1 x3\<rbrakk> \<Longrightarrow> P1 (LET x1 = x2 IN x3)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P2 (x1, x2)"   and "P3 []"   and "\<And>x1 x2. \<lbrakk>P2 x1; P3 x2\<rbrakk> \<Longrightarrow> P3 (x1 \<Colon> x2)" shows "P3 char_list_trm_prod_list"
cet_add_int: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e1 :: "cexpr"   and e2 :: "cexpr" assumes "\<Gamma> \<turnstile>\<^sub>c e1 : INTEG"   and "\<Gamma> \<turnstile>\<^sub>c e2 : INTEG" shows "\<Gamma> \<turnstile>\<^sub>c e1 +\<^sub>c e2 : INTEG"
assms_2: shows "f = (\<lambda>y. {z. (y, z) \<in> S})"
vec_last_carrier: fixes v :: "'b Matrix.vec"   and n :: "nat" shows "vec_last v n \<in> carrier_vec n"
rel_fun_conversep: fixes A :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_fun A\<inverse>\<inverse> B\<inverse>\<inverse> = (rel_fun A B)\<inverse>\<inverse>"
mult_mult_of: fixes R :: "('c, 'd) ring_scheme" shows "(\<otimes>\<^bsub>Ring_Divisibility.mult_of R\<^esub>) = (\<otimes>\<^bsub>R\<^esub>)"
ord_isom_func: fixes E :: "('g, 'h) Order_scheme"   and f :: "'a \<Rightarrow> 'g" assumes "Order E"   and "ord_isom D E f" shows "f \<in> carrier D \<rightarrow> carrier E"
ord_isom_func: fixes E :: "('e, 'f) Order_scheme"   and f :: "'a \<Rightarrow> 'e" assumes "Order E"   and "ord_isom D E f" shows "f \<in> carrier D \<rightarrow> carrier E"
L_estimate: shows "\<bar>\<Sum>i. L (i + Suc (max N_L N_f'))\<bar> < r / 3"
Par1F: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi"   and Q :: "pi" assumes "P \<Longrightarrow>\<^sub>l \<alpha> \<prec> P'" shows "P \<parallel> Q \<Longrightarrow>\<^sub>l \<alpha> \<prec> P' \<parallel> Q"
map_idI: fixes xs :: "'d list"   and f :: "'d \<Rightarrow> 'd" assumes "\<And>x. x \<in> set xs \<Longrightarrow> f x = x" shows "map f xs = xs"
A3: fixes xs :: "'a list"   and ys :: "'a list" assumes "xs \<noteq> []"   and "ys \<noteq> []"   and "P xs"   and "P ys" shows "P (xs @ ys)"