hcju/isabelleps huggingface dataset:
odeliverT: fixes \<sigma>' :: "nat \<Rightarrow> 'e"   and i :: "nat"   and \<sigma> :: "nat \<Rightarrow> 'e"   and l :: "'h"   and s\<^sub>d\<^sub>a\<^sub>t\<^sub>a :: "'e \<Rightarrow> nat"   and p :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" assumes "\<sigma>' i = \<sigma> i" shows "((\<sigma>, {l}deliver(s\<^sub>d\<^sub>a\<^sub>t\<^sub>a) . p), deliver (s\<^sub>d\<^sub>a\<^sub>t\<^sub>a (\<sigma> i)), \<sigma>', p) \<in> oseqp_sos \<Gamma> i"
matchs_induct: fixes x4 :: "(char list \<times> pat) list"   and x5 :: "(char list \<times> trm) list"   and x6 :: "trm list"   and P2 :: "(char list \<times> pat) list \<Rightarrow> (char list \<times> trm) list \<Rightarrow> trm list \<Rightarrow> bool" assumes "\<turnstile> x4 [\<rhd>] x5 \<Rightarrow> x6"   and "\<And>fs. P2 [] fs []"   and "\<And>fs l t p ts fps us. \<lbrakk>fs\<langle>l\<rangle>\<^sub>? = \<lfloor>t\<rfloor>; \<turnstile> p \<rhd> t \<Rightarrow> ts; \<turnstile> fps [\<rhd>] fs \<Rightarrow> us; P2 fps fs us\<rbrakk> \<Longrightarrow> P2 ((l, p) \<Colon> fps) fs (ts @ us)" shows "P2 x4 x5 x6"
min_height_le_height: fixes kdt :: "'a kdt" shows "min_height kdt \<le> height kdt"
subst_cls_mset_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and CC :: "'b literal multiset multiset"   and \<sigma> :: "'c" shows "substitution_ops.subst_cls_mset subst_atm CC \<sigma> = {#substitution_ops.subst_cls subst_atm A \<sigma>. A \<in># CC#}"
three_composable_arrows_of_spans_axioms: shows "three_composable_arrows_of_spans (\<cdot>) prj0 prj1 (cod \<mu>) (cod \<nu>) (cod \<pi>)"
reduction_doesnt_break_subsets: fixes p :: "real poly"   and subsets :: "nat list list"   and qs :: "real poly list"   and signs :: "rat list list" assumes "p \<noteq> 0"   and "all_list_constr subsets (length qs)"   and "satisfy_equation p qs subsets signs"   and "invertible_mat (M_mat signs subsets)" shows "all_list_constr (reduction_subsets p qs signs subsets (M_mat signs subsets)) (length qs)"
w_simps_4: shows "trg \<chi>.chine = src u\<^sub>0"
double_not_eq_Suc_double: fixes m :: "nat"   and n :: "nat" shows "2 * m \<noteq> Suc (2 * n)"
Lxy_distinct: fixes xs :: "'c list"   and S :: "'c set" assumes "distinct xs" shows "distinct (Lxy xs S)"
bs_set_rel_sv: shows "single_valued (\<langle>nat_rel\<rangle>bs_set_rel)"
eq_refl: fixes x :: "'a"   and y :: "'a" assumes "x = y" shows "domain_semigroup.ds_ord (antidomain_left_monoid.am_d ar) (\<lambda>x y. y \<cdot> x) x y"
rset_error_code_def: fixes s :: "rstate_t"   and a :: "int_point_t" shows "rset_error_code s a \<equiv> s"
accept_sim: fixes w :: "nat \<Rightarrow> 'l" assumes "b.accept w" shows "a.accept w"
Suc: shows "remaining_qty Mkt v pf asset (Suc n) \<in> borel_measurable (F n)"
simps_1: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "linorder.msort less [] = []"
order_0I: fixes p :: "'c poly"   and a :: "'c" assumes "poly p a \<noteq> (0::'c)" shows "order a p = 0"
bound_on_all_plans_bounds_problem_plan_bound_charles_i: fixes PROB :: "(('c, bool) fmap \<times> ('c, bool) fmap) set"   and k :: "nat" assumes "finite PROB"   and "\<forall>x\<in>MPLS_charles PROB. x < k" shows "Sup (MPLS_charles PROB) \<in> MPLS_charles PROB"
is_continuation_def: fixes k :: "nat"   and u :: "nat"   and q :: "(symbol \<times> character list) list"   and ts :: "(symbol \<times> character list) list" shows "is_continuation k u q ts = (q \<in> \<P> k u \<and> charslength q = k \<and> admissible (q @ ts) \<and> (\<forall>t\<in>set ts. t \<in> \<Z> k (Suc u)) \<and> (\<forall>t\<in>set (butlast ts). chars_of_token t = []))"
Y: shows "\<forall>X\<in>A. X \<subseteq># Y"
sigma_finite_measure: fixes M :: "'a measure" assumes "finite_measure M" shows "sigma_finite_measure M"
simps_3: fixes n0 :: "nat"   and j0 :: "nat"   and i0 :: "nat"   and n1 :: "nat"   and j1 :: "nat"   and ix1 :: "nat"   and L :: "(nat \<times> nat \<times> nat) list" shows "ladder_stepdown ((n0, j0, i0) # (n1, j1, ix1) # L) = (n1 - Suc n0, j1, j0 + ix1) # ladder_shift_n (Suc n0) L"
lm060: fixes G :: "'d set"   and N :: "'c set" shows "allAllocations N G \<subseteq> {a. \<Union> (Range a) = G}"
L_transition_eqvt: fixes P\<^sub>L :: "('state, 'effect) L_state"   and \<alpha>\<^sub>LP\<^sub>L' :: "(('act, 'effect) L_action, ('state, 'effect) L_state) residual"   and p :: "perm" assumes "P\<^sub>L \<rightarrow>\<^sub>L \<alpha>\<^sub>LP\<^sub>L'" shows "p \<bullet> P\<^sub>L \<rightarrow>\<^sub>L p \<bullet> \<alpha>\<^sub>LP\<^sub>L'"
NSLIM_const_eq: fixes k :: "'d"   and a :: "'c"   and l :: "'d" assumes "(\<lambda>x. k) \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S l" shows "k = l"
bin_cat_inj: fixes n :: "nat"   and b :: "int"   and a :: "int"   and d :: "int"   and c :: "int" shows "(concat_bit n b a = concat_bit n d c) = (a = c \<and> take_bit n b = take_bit n d)"
elims: fixes x :: "ternaryvalue"   and xa :: "ternaryvalue"   and y :: "ternaryvalue" assumes "eval_ternary_And x xa = y" obtains "x = TernaryTrue" and "xa = TernaryTrue" and "y = TernaryTrue"   | "x = TernaryTrue" and "xa = TernaryFalse" and "y = TernaryFalse"   | "x = TernaryFalse" and "xa = TernaryTrue" and "y = TernaryFalse"   | "x = TernaryFalse" and "xa = TernaryFalse" and "y = TernaryFalse"   | "x = TernaryFalse" and "xa = TernaryUnknown" and "y = TernaryFalse"   | "x = TernaryTrue" and "xa = TernaryUnknown" and "y = TernaryUnknown"   | "x = TernaryUnknown" and "xa = TernaryFalse" and "y = TernaryFalse"   | "x = TernaryUnknown" and "xa = TernaryTrue" and "y = TernaryUnknown"   | "x = TernaryUnknown" and "xa = TernaryUnknown" and "y = TernaryUnknown"
det_permute_rows: fixes A :: "'b mat"   and n :: "nat"   and p :: "nat \<Rightarrow> nat" assumes "A \<in> carrier_mat n n"   and "p permutes {0..<n}" shows "det (mat n n (\<lambda>(i, j). A $$ (p i, j))) = signof p * det A"
monoD: fixes F :: "ordinal \<Rightarrow> ordinal"   and x :: "ordinal"   and y :: "ordinal" assumes "normal F"   and "x \<le> y" shows "F x \<le> F y"
Suc_4: shows "u \<in> gusuccessors A w v"
elims_2: fixes x :: "('a, 'b) Term.term"   and xa :: "('a, 'b) Term.term" assumes "subterm_of x xa" obtains t :: "('a, 'b) Term.term" and y :: "'b" where "x = ta__" and "xa = Var y" and "ta__ = Var y"   | t :: "('a, 'b) Term.term" and f :: "'a" and T :: "('a, 'b) Term.term list" where "x = t" and "xa = Fun f T" and "t = Fun f T \<or> list_ex (subterm_of t) T"
simps_4: fixes b :: "nat"   and \<phi> :: "'b formula"   and \<psi> :: "'b formula" shows "fvi b (Or \<phi> \<psi>) = fvi b \<phi> \<union> fvi b \<psi>"
MemReturn_def: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rs :: "state \<Rightarrow> PrIds \<Rightarrow> Vals"   and p :: "PrIds" shows "MemReturn ch rs p \<equiv> $(rs!p) \<noteq> #NotAResult \<and> (rs!p)$ = #NotAResult \<and> Return ch p (rs!p)"
distinct_3: fixes x31 :: "'a"   and x32 :: "'a"   and x33 :: "'b" shows "DenyAll \<noteq> AllowPortFromTo x31 x32 x33"
al: shows "\<phi> (t' (n - \<langle>c \<and> t\<rangle> - 1))"
disc_2: fixes x21 :: "char list"   and x22 :: "'a list" shows "\<not> sign (Neg x21 x22)"
Ass_lemma: fixes G :: "prog"   and s0 :: "st"   and var :: "Term.var"   and w :: "val"   and f :: "val \<Rightarrow> state \<Rightarrow> state"   and s1 :: "st"   and e :: "expr"   and v :: "val"   and s2 :: "st"   and eT :: "ty"   and L :: "lenv" assumes "G\<turnstile>Norm s0 \<midarrow>var=\<succ>(w, f)\<rightarrow> Norm s1"   and "G\<turnstile>Norm s1 \<midarrow>e-\<succ>v\<rightarrow> Norm s2"   and "G,s2\<turnstile>v\<Colon>\<preceq>eT"   and "s1\<le>|s2 \<longrightarrow> assign f v (Norm s2)\<Colon>\<preceq>(G, L)" shows "assign f v (Norm s2)\<Colon>\<preceq>(G, L) \<and> (normal (assign f v (Norm s2)) \<longrightarrow> G,snd (assign f v (Norm s2))\<turnstile>v\<Colon>\<preceq>eT)"
assms_2: fixes x :: "'b" assumes "nofail m"   and "inres m x"   and "f' x \<le> SPEC \<Psi>" shows "f x \<le> SPEC \<Phi>"
release_and_right_distrib: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi>\<^sub>1 :: "'b ltln"   and \<psi>\<^sub>2 :: "'b ltln" shows "w \<Turnstile>\<^sub>n \<phi> R\<^sub>n (\<psi>\<^sub>1 and\<^sub>n \<psi>\<^sub>2) = w \<Turnstile>\<^sub>n \<phi> R\<^sub>n \<psi>\<^sub>1 and\<^sub>n \<phi> R\<^sub>n \<psi>\<^sub>2"
matricize_def: fixes rmodes :: "nat set"   and T :: "'a tensor" shows "matricize rmodes T = mat (prod_list (nths (dims T) rmodes)) (prod_list (nths (dims T) (- rmodes))) (\<lambda>(r, c). lookup T (weave rmodes (digit_encode (nths (dims T) rmodes) r) (digit_encode (nths (dims T) (- rmodes)) c)))"
intro: fixes wtFsym :: "'b \<Rightarrow> bool"   and wtPsym :: "'c \<Rightarrow> bool" assumes "Sig.Signature TYPE('a) wtFsym wtPsym" shows "M_Signature TYPE('a) wtFsym wtPsym"
assms_4: fixes A :: "V"   and B :: "V" assumes "A \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"   and "B \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"   and "A \<in>\<^sub>\<circ> Vset \<alpha>"   and "B \<in>\<^sub>\<circ> Vset \<alpha>" shows "\<Union>\<^sub>\<circ> ((\<lambda>a\<in>\<^sub>\<circ>A. \<Union>\<^sub>\<circ> (VLambda B (Hom \<CC> a) `\<^sub>\<circ> B)) `\<^sub>\<circ> A) \<in>\<^sub>\<circ> Vset \<alpha>"
eventually_nonzero_ident_nhds: fixes a :: "'a" shows "eventually_nonzero (nhds a) (\<lambda>x. x) = (a \<noteq> (0::'a))"
assms: shows "t \<in> subst_range (\<theta>1 \<circ>\<^sub>s \<theta>2)"
Or: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and j :: "nat"   and n :: "nat"   and R :: "'a list set"   and \<phi> :: "'a mformula"   and \<phi>' :: "'a MFOTL.formula"   and \<psi> :: "'a mformula"   and \<psi>' :: "'a MFOTL.formula"   and buf :: "'a option list set list \<times> 'a option list set list" assumes "wf_mformula \<sigma> j n R \<phi> \<phi>'"   and "wf_mformula \<sigma> j n R \<psi> \<psi>'"   and "fv \<phi>' = fv \<psi>'"   and "wf_mbuf2' \<sigma> j n R \<phi>' \<psi>' buf" shows "wf_mformula \<sigma> j n R (MOr \<phi> \<psi> buf) (formula.Or \<phi>' \<psi>')"
pdevs_apply_pdevs_of_list_Nil: fixes i :: "nat" shows "pdevs_apply (pdevs_of_list []) i = (0::'a)"
snoc_2: shows "crdt_ops (xs @ [(oid, oper)] @ ys @ [y]) deps"
mult_cancel_left2: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and c :: "'d"   and a :: "'d" assumes "class.ring_1_no_zero_divisors one times plus zero minus uminus" shows "(times c a = c) = (c = zero \<or> a = one)"
F_def: shows "F \<equiv> \<lambda>x. pmf p x * measure_pmf.prob (f x) A / measure_pmf.prob (p \<bind> f) A"
residue_plus_zero_l: fixes k :: "nat"   and y :: "int" shows "y \<oplus>\<^bsub>residue_ring (p ^ k)\<^esub> 0 = y mod p ^ k"
pw_THROW_1: fixes e :: "'c" shows "nofail (THROW e)"
widen_null: fixes P :: "'b prog"   and C :: "char list" shows "subtype P NT (Class C)"
arcs_distinct_dest1: fixes i :: "nat"   and a :: "nat"   and zs :: "nat list"   and b :: "nat"   and c :: "nat"   and j :: "nat" assumes "distinct (i # a # zs)"   and "(b, c) \<in> set (arcs a j zs)" shows "b \<noteq> i"
below_eq_trans: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "below a b"   and "b = c" shows "below a c"
supp_list_cons: fixes x :: "'e"   and xs :: "'e list" shows "supp (x # xs) = supp x \<union> supp xs"
asmp: shows "(n', nid') = (SOME x. (\<exists>n. latestAct_cond nid'' t n'' n) \<and> \<parallel>snd x\<parallel>\<^bsub>t (fst x)\<^esub> \<and> fst x = \<langle>nid'' \<leftarrow> t\<rangle>\<^bsub>n''\<^esub> \<and> (prefix (bc (\<sigma>\<^bsub>nid''\<^esub>t n'')) (bc (\<sigma>\<^bsub>snd x\<^esub>t (fst x))) \<or> (\<exists>b. bc (\<sigma>\<^bsub>nid''\<^esub>t n'') = bc (\<sigma>\<^bsub>snd x\<^esub>t (fst x)) @ [b] \<and> mining (\<sigma>\<^bsub>nid''\<^esub>t n''))))"
TryRed: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and V :: "char list"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow> \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>try e catch(C V) e\<^sub>2,s,b\<rangle> \<rightarrow> \<langle>try e' catch(C V) e\<^sub>2,s',b'\<rangle>"
id_1: shows "berlekamp_basis f = [] @ berlekamp_basis f"
induct: fixes P :: "'c \<Rightarrow> ('c \<times> 'd) tree \<Rightarrow> bool"   and a0 :: "'c"   and a1 :: "('c \<times> 'd) tree" assumes "\<And>x. P x \<langle>\<rangle>"   and "\<And>x a t1 t2. x = fst a \<Longrightarrow> P x \<langle>t1, a, t2\<rangle>"   and "\<And>x a b t1 t2 t3. \<lbrakk>x = fst a; x < fst b\<rbrakk> \<Longrightarrow> P x \<langle>\<langle>t1, a, t2\<rangle>, b, t3\<rangle>"   and "\<And>x a t. x < fst a \<Longrightarrow> P x \<langle>\<langle>\<rangle>, a, t\<rangle>"   and "\<And>x a b t1 t2. \<lbrakk>x < fst a; x < fst b\<rbrakk> \<Longrightarrow> P x \<langle>\<langle>\<langle>\<rangle>, a, t1\<rangle>, b, t2\<rangle>"   and "\<And>x a b t1 t2 t3. \<lbrakk>x < fst a; x < fst b; t1 \<noteq> \<langle>\<rangle>; P x t1\<rbrakk> \<Longrightarrow> P x \<langle>\<langle>t1, a, t2\<rangle>, b, t3\<rangle>"   and "\<And>a x b t1 t2. \<lbrakk>fst a < x; x < fst b\<rbrakk> \<Longrightarrow> P x \<langle>\<langle>t1, a, \<langle>\<rangle>\<rangle>, b, t2\<rangle>"   and "\<And>a x b t2 t1 t3. \<lbrakk>fst a < x; x < fst b; t2 \<noteq> \<langle>\<rangle>; P x t2\<rbrakk> \<Longrightarrow> P x \<langle>\<langle>t1, a, t2\<rangle>, b, t3\<rangle>"   and "\<And>a x b t1 t2 t3. \<lbrakk>fst a < x; x = fst b\<rbrakk> \<Longrightarrow> P x \<langle>t1, a, \<langle>t2, b, t3\<rangle>\<rangle>"   and "\<And>a x t. fst a < x \<Longrightarrow> P x \<langle>t, a, \<langle>\<rangle>\<rangle>"   and "\<And>a x b t2 t1 t3. \<lbrakk>fst a < x; x < fst b; t2 \<noteq> \<langle>\<rangle>; P x t2\<rbrakk> \<Longrightarrow> P x \<langle>t1, a, \<langle>t2, b, t3\<rangle>\<rangle>"   and "\<And>a x b t1 t2. \<lbrakk>fst a < x; x < fst b\<rbrakk> \<Longrightarrow> P x \<langle>t1, a, \<langle>\<langle>\<rangle>, b, t2\<rangle>\<rangle>"   and "\<And>a x b t1 t2. \<lbrakk>fst a < x; fst b < x\<rbrakk> \<Longrightarrow> P x \<langle>t1, a, \<langle>t2, b, \<langle>\<rangle>\<rangle>\<rangle>"   and "\<And>a x b t3 t1 t2. \<lbrakk>fst a < x; fst b < x; t3 \<noteq> \<langle>\<rangle>; P x t3\<rbrakk> \<Longrightarrow> P x \<langle>t1, a, \<langle>t2, b, t3\<rangle>\<rangle>" shows "P a0 a1"
INF_greatest: fixes Inf :: "'a::type set \<Rightarrow> 'a::type"   and Sup :: "'a::type set \<Rightarrow> 'a::type"   and inf :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type"   and less_eq :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and less :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and sup :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type"   and bot :: "'a::type"   and top :: "'a::type"   and A :: "'b::type set"   and u :: "'a::type"   and f :: "'b::type \<Rightarrow> 'a::type" assumes "class.complete_lattice (Inf::'a::type set \<Rightarrow> 'a::type) (Sup::'a::type set \<Rightarrow> 'a::type) (inf::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (less::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (sup::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (bot::'a::type) (top::'a::type)"   and "\<And>i::'b::type. i \<in> (A::'b::type set) \<Longrightarrow> (less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (u::'a::type) ((f::'b::type \<Rightarrow> 'a::type) i)" shows "(less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (u::'a::type) ((Inf::'a::type set \<Rightarrow> 'a::type) ((f::'b::type \<Rightarrow> 'a::type) ` (A::'b::type set)))"
normalize_andfold_MatchExp_normalized_n_primitive: fixes ms :: "'b match_expr list"   and disc :: "'b \<Rightarrow> bool"   and sel :: "'b \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> bool"   and s :: "'b match_expr" assumes "\<forall>m\<in>set ms. \<forall>s'\<in>set (normalize_match m). normalized_n_primitive (disc, sel) f s'"   and "s \<in> set (normalize_match (andfold_MatchExp ms))" shows "normalized_n_primitive (disc, sel) f s"
rel_compp_Grp: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_id R = (BNF_Def.Grp {x. set_id x \<subseteq> {(x, y). R x y}} (map_id fst))\<inverse>\<inverse> OO BNF_Def.Grp {x. set_id x \<subseteq> {(x, y). R x y}} (map_id snd)"
wqo_on_hom: fixes Q :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'a"   and A :: "'c set"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "transp_on Q (h ` A)"   and "\<forall>x\<in>A. \<forall>y\<in>A. P x y \<longrightarrow> Q (h x) (h y)"   and "wqo_on P A" shows "wqo_on Q (h ` A)"
intro: fixes \<Gamma> :: "'b inference set" assumes "reductive_inference_system \<Gamma>" shows "standard_redundancy_criterion_reductive \<Gamma>"
sup_union_left1: fixes x :: "'b"   and N :: "'b multiset"   and M :: "'b multiset" assumes "x \<notin># N" shows "add_mset x M \<union># N = add_mset x (M \<union># N)"
dual_le: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "less_eq x y = less_eq (dual y) (dual x)"
coprime_linear_poly': fixes c :: "'a"   and c' :: "'a" assumes "c \<noteq> c'"   and "c \<noteq> (0::'a)"   and "c' \<noteq> (0::'a)" shows "coprime [:1::'a, c:] [:1::'a, c':]"
seq_def: fixes P :: "'a process"   and Q :: "'a process" shows "P `;` Q \<equiv> Abs_process ({(t, X). (t, X \<union> {tick}) \<in> F P \<and> tickFree t} \<union> {(t, X). \<exists>t1 t2. t = t1 @ t2 \<and> t1 @ [tick] \<in> T P \<and> (t2, X) \<in> F Q} \<union> {(t, X). t \<in> div_seq P Q}, div_seq P Q)"
wls_obtain_rep: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and x' :: "'var"   and x :: "'var" assumes "wls s X"   and "fresh xs x' X" shows "\<exists>X'. skel X' = skel X \<and> (X, X') \<in> swapped \<and> wls s X' \<and> Abs xs x X = Abs xs x' X'"
fields_emptyI: fixes G :: "prog"   and C :: "qtname"   and c :: "class"   and y :: "class" assumes "ws_prog G"   and "class G C = Some c"   and "cfields c = []"   and "C \<noteq> Object \<longrightarrow> class G (super c) = Some y \<and> DeclConcepts.fields G (super c) = []" shows "DeclConcepts.fields G C = []"
above_Lin: fixes x :: "alt"   and y :: "alt"   and L :: "(alt \<times> alt) set" assumes "x \<noteq> y"   and "L \<in> Lin" shows "Arrow_Order.above L x y \<in> Lin"
a_less_b: fixes i :: "'a" assumes "i \<in> Basis" shows "a \<bullet> i < b \<bullet> i"
sestoftUnGCstep: fixes r :: "Vars.var list"   and c :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and d :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" assumes "to_gc_conf r c \<Rightarrow>\<^sub>G d"   and "heap_upds_ok_conf c"   and "closed c"   and "r_ok r c" shows "\<exists>r' c'. c \<Rightarrow>\<^sup>* c' \<and> d = to_gc_conf r' c' \<and> r_ok r' c'"
vector_diff_chain_at: fixes f :: "real \<Rightarrow> real"   and f' :: "real"   and x :: "real"   and g :: "real \<Rightarrow> 'a"   and g' :: "'a" assumes "(f has_vector_derivative f') (at x)"   and "(g has_vector_derivative g') (at (f x))" shows "(g \<circ> f has_vector_derivative f' *\<^sub>R g') (at x)"
pref_exps_pow: fixes k :: "nat"   and l :: "nat"   and r :: "'b list" assumes "k \<le> l" shows "r \<^sup>@ k \<le>p r \<^sup>@ l"
tpOfV_getVars_nth: fixes i :: "nat"   and \<sigma>l :: "'tp list" assumes "i < length \<sigma>l" shows "GE.tpOfV (GE.getVars \<sigma>l ! i) = \<sigma>l ! i"
hom_char: fixes a :: "'a"   and b :: "'a" shows "Left.hom a b = hom a b \<inter> Collect (left (trg f))"
eval_in_hom': fixes t :: "'c local.term" assumes "Arr t" shows "\<guillemotleft>\<lbrace>t\<rbrace> : \<lbrace>Src t\<rbrace> \<rightarrow> \<lbrace>Trg t\<rbrace>\<guillemotright> \<and> \<guillemotleft>\<lbrace>t\<rbrace> : \<lbrace>Dom t\<rbrace> \<Rightarrow> \<lbrace>Cod t\<rbrace>\<guillemotright>"
compact_imp_closed: fixes "open" :: "'a set \<Rightarrow> bool"   and s :: "'a set" assumes "class.t2_space open"   and "topological_space.compact open s" shows "topological_space.closed open s"
vfsequence_vdomain: shows "\<D>\<^sub>\<circ> S = vcard S"
of_int_1: fixes one :: "'q"   and times :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and plus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and zero :: "'q"   and minus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and uminus :: "'q \<Rightarrow> 'q" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus 1 = one"
apply_commute: fixes x :: "('b, 'c) commute" shows "apply_commute x \<in> {f. \<forall>x y. f x y = f y x}"
interleaving_Nil_implies_eq1: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "interleavings xs ys zs"   and "xs = []" shows "ys = zs"
minus_equality: fixes G :: "('a, 'b) ring_scheme"   and y :: "'a"   and x :: "'a" assumes "abelian_group G"   and "y \<oplus>\<^bsub>G\<^esub> x = \<zero>\<^bsub>G\<^esub>"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "\<ominus>\<^bsub>G\<^esub> x = y"
filter_singleton: fixes f :: "'c \<Rightarrow> bool"   and s :: "'c list"   and y :: "'c" assumes "filter f s = [y]" shows "f y \<and> y \<in> set s"
continuous_on_codomain: fixes c :: "('b, 'a) chart" shows "continuous_on (codomain c) (inv_chart c)"
bernoulli_altdef: fixes n :: "nat" shows "bernoulli n = (\<Sum>m\<le>n. \<Sum>k\<le>m. (- 1) ^ k * real (m choose k) * real k ^ n / real (Suc m))"
C_case_Cinf: fixes f :: "C \<rightarrow> 'b" shows "C_case\<cdot>f\<cdot>C\<^sup>\<infinity> = f\<cdot>C\<^sup>\<infinity>"
lm144: fixes A :: "'a set" shows "infinite A = (card (Pow A) = 0)"
intro: fixes G :: "('c, 'd) monoid_scheme" assumes "\<And>a. \<lbrakk>a \<in> carrier G; irreducible G a\<rbrakk> \<Longrightarrow> prime G a" shows "primeness_condition_monoid_axioms G"
lower_character_lift_character_2: fixes \<chi> :: "'a \<Rightarrow> complex"   and z :: "complex" assumes "\<chi> \<in> characters (G\<lparr>carrier := {\<one>}\<rparr>)" shows "H.lift_character (\<chi>, z) x = z"
rotate5: fixes A5 :: "fmla"   and A1 :: "fmla"   and A2 :: "fmla"   and A3 :: "fmla"   and A4 :: "fmla"   and H :: "fmla set"   and B :: "fmla" assumes "insert A5 (insert A1 (insert A2 (insert A3 (insert A4 H)))) \<turnstile> B" shows "insert A1 (insert A2 (insert A3 (insert A4 (insert A5 H)))) \<turnstile> B"
Restrp_transitive: shows "transitive UNIV ((\<sqsubseteq>) \<restriction> A \<restriction> UNIV)"
openin_open_Int: fixes S :: "'b set"   and U :: "'b set" assumes "open S" shows "openin (top_of_set U) (U \<inter> S)"
content_nonneg: fixes a :: "'a"   and b :: "'a" shows "\<not> content (cbox a b) < 0"
nvC_tC: fixes \<sigma> :: "'tp"   and c :: "('fsym, 'a) lit list" assumes "\<not> unprot \<sigma>" shows "\<forall>x\<in>nvC (tC c). TE.tpOfV x \<noteq> \<sigma>"
disj_assoc: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and S :: "'b \<Rightarrow> bool" shows "((P \<or> Q) \<or> S) = (P \<or> Q \<or> S)"
D': fixes \<sigma>' :: "nat list"   and x :: "nat"   and y :: "nat"   and h :: "nat list"   and h0 :: "nat list" assumes "\<sigma>' \<in> Lxx x y"   and "x \<noteq> y"   and "TS_inv' ([x, y], h) x [x, y]" shows "Partial_Cost_Model.T_on' (rTS h0) ([x, y], h) \<sigma>' \<le> 2 * T\<^sub>p [x, y] \<sigma>' (OPT2 \<sigma>' [x, y]) \<and> TS_inv (Partial_Cost_Model.config'_rand (Partial_Cost_Model.embed (rTS h0)) (return_pmf ([x, y], h)) \<sigma>') (last \<sigma>') [x, y]"
sint_of_nat_ge_zero: fixes x :: "nat" assumes "x < 2 ^ (LENGTH('b) - 1)" shows "0 \<le> sint (word_of_nat x)"