hcju/isabelleps huggingface dataset:
NextMem_def: fixes SR :: "(('e \<times> ('c \<Rightarrow> 'd)) \<times> 'e list \<times> 'e option \<times> ('c \<Rightarrow> 'd)) set"   and c :: "'e"   and m :: "'c \<Rightarrow> 'd" shows "WHATWHERE.NextMem SR c m \<equiv> THE m'. \<exists>p \<alpha>. ((c, m), \<alpha>, p, m') \<in> SR"
has_sort_sort_leq: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and T :: "typ"   and S :: "String.literal set"   and S' :: "String.literal set" assumes "wf_theory \<Theta>"   and "has_sort (osig (sig \<Theta>)) T S"   and "sort_leq (subclass (osig (sig \<Theta>))) S S'" shows "has_sort (osig (sig \<Theta>)) T S'"
induct: fixes x1 :: "'b com option list"   and x2 :: "'b"   and x3 :: "'b com option list"   and x4 :: "'b"   and P :: "'b com option list \<Rightarrow> 'b \<Rightarrow> 'b com option list \<Rightarrow> 'b \<Rightarrow> bool" assumes "(x1, x2) -pc\<rightarrow> (x3, x4)"   and "\<And>i Ps s r t. \<lbrakk>i < length Ps; (Ps ! i, s) -c\<rightarrow> (r, t)\<rbrakk> \<Longrightarrow> P Ps s (Ps[i := r]) t" shows "P x1 x2 x3 x4"
assms_2: shows "\<forall>a\<in>A. bo a u"
assms_6: fixes zs :: "('a \<times> 'c) list" assumes "oalist_inv_raw zs" shows "h zs = map_val_pair (\<lambda>k. f k (0::'b)) zs"
strict_prefixI': fixes ys :: "'a list"   and xs :: "'a list"   and z :: "'a"   and zs :: "'a list" assumes "ys = xs @ z # zs" shows "strict_prefix xs ys"
oseq_step_invariant_sterms: fixes A :: "((nat \<Rightarrow> 'a) \<times> ('a, 'b, 'c, 'd) seqp, 'b seq_action) automaton"   and S :: "(nat \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> 'b seq_action \<Rightarrow> bool"   and U :: "(nat \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> bool"   and \<Gamma> :: "'c \<Rightarrow> ('a, 'b, 'c, 'd) seqp"   and P :: "((nat \<Rightarrow> 'a) \<times> 'd) \<times> 'b seq_action \<times> (nat \<Rightarrow> 'a) \<times> 'd \<Rightarrow> bool"   and i :: "nat"   and l' :: "'d"   and q :: "('a, 'b, 'c, 'd) seqp"   and \<sigma> :: "nat \<Rightarrow> 'a"   and p :: "('a, 'b, 'c, 'd) seqp"   and p' :: "('a, 'b, 'c, 'd) seqp"   and a :: "'b seq_action"   and \<sigma>' :: "nat \<Rightarrow> 'a" assumes "A \<Turnstile>\<^sub>A (S, U \<rightarrow>) onll \<Gamma> P"   and "wellformed \<Gamma>"   and "automaton.trans A = oseqp_sos \<Gamma> i"   and "l' \<in> labels \<Gamma> q"   and "(\<sigma>, p) \<in> oreachable A S U"   and "((\<sigma>, p'), a, \<sigma>', q) \<in> automaton.trans A"   and "S \<sigma> \<sigma>' a"   and "p' \<in> sterms \<Gamma> p" shows "\<forall>l\<in>labels \<Gamma> p'. P ((\<sigma>, l), a, \<sigma>', l')"
Push_neq_K0: fixes k :: "nat"   and f :: "nat \<Rightarrow> 'a + nat"   and P :: "bool" assumes "Push (Inr (Suc k)) f = (\<lambda>z. Inr 0)" shows "P"
alwaysD: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b"   and i :: "nat" assumes "(\<box>P) \<sigma>" shows "P (\<sigma> |\<^sub>s i)"
simps_36: fixes x3 :: "'a" shows "set3_sum3 (In3 x3) = {x3}"
reindex_bij_betw: fixes h :: "'b \<Rightarrow> 'c"   and S :: "'b set"   and T :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "bij_betw h S T" shows "(\<Prod>x\<in>S. g (h x)) = prod g T"
Ord_mem_succ_cases: fixes k :: "hf"   and l :: "hf" assumes "Ord k"   and "l \<^bold>\<in> k" shows "succ l = k \<or> succ l \<^bold>\<in> k"
seq_char: fixes g :: "'c local.term set"   and f :: "'c local.term set" shows "seq g f = (g \<cdot> f \<noteq> null)"
siso_cont_indis: fixes c :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and t :: "'state"   and i :: "nat" assumes "siso c"   and "s \<approx> t"   and "i < brn c" shows "eff c s i \<approx> eff c t i \<and> wt c s i = wt c t i \<and> local.cont c s i = local.cont c t i"
SN_rel_on_imp_SN_on: fixes R :: "('a \<times> 'a) set"   and S :: "('a \<times> 'a) set"   and T :: "'a set" assumes "SN_rel_on R S T" shows "SN_on R T"
adds: fixes f :: "'t \<Rightarrow>\<^sub>0 'b" assumes "f \<in> pmdl F"   and "f \<in> dgrad_p_set d m"   and "f \<noteq> 0" shows "\<exists>g\<in>G. g \<noteq> 0 \<and> lt g adds\<^sub>t lt f"
divergence_refine_def: fixes P :: "'b process"   and Q :: "'b process" shows "P \<sqsubseteq>\<^sub>D Q \<equiv> D Q \<subseteq> D P"
rcs_fixed: fixes G :: "('c, 'd) Group_scheme"   and H :: "'c set"   and a :: "'c" assumes "Group G"   and "G \<guillemotright> H "   and "a \<in> carrier G"   and "H \<bullet>\<^bsub>G\<^esub> a = H" shows "a \<in> H"
zero_le_divide_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'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 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 a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq zero (divide a b) = (less_eq zero a \<and> less_eq zero b \<or> less_eq a zero \<and> less_eq b zero)"
BinOpThrow2: fixes v\<^sub>1 :: "val"   and bop :: "bop"   and e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" shows "P \<turnstile> \<langle>Val v\<^sub>1 \<guillemotleft>bop\<guillemotright> throw e,s\<rangle> \<rightarrow> \<langle>throw e,s\<rangle>"
measure_pmf_inject: fixes x :: "'b pmf"   and y :: "'b pmf" shows "(measure_pmf x = measure_pmf y) = (x = y)"
mult_isor: fixes x :: "'a list set"   and y :: "'a list set"   and z :: "'a list set" assumes "x \<subseteq> y" shows "p_prod x z \<subseteq> p_prod y z"
Ext_subset_mono: fixes R :: "('a \<times> 'b) set"   and S :: "('a \<times> 'b) set"   and y :: "'b" assumes "R \<subseteq> S" shows "Ext R y \<subseteq> Ext S y"
assms: shows "(f, g) \<in> \<langle>Id\<rangle>nres_rel"
rb_aux_inv_init: shows "rb_aux_inv ([], Koszul_syz_sigs fs, map Inr [0..<length fs])"
axioms_2: fixes G :: "'b pair_pre_digraph" assumes "pair_graph G" shows "pair_pseudo_graph G"
Sup_pres_Inf_pres: fixes f :: "'c \<Rightarrow> 'd" shows "Sup_pres f = Inf_pres (\<partial>\<^sub>F f)"
separable_def: fixes k :: "nat" shows "separable k = (gauss_sum k = cnj (\<chi> k) * gauss_sum 1)"
one_dim_iso_def: fixes a :: "'c" shows "one_dim_iso a = of_complex \<langle>1::'c, a\<rangle>"
map_simp_ide: fixes a :: "'a" assumes "ide a" shows "\<rho>.map a = pr1 a (cod \<iota>)"
word_div_less: fixes m :: "'a word"   and n :: "'a word" assumes "m < n" shows "m div n = 0"
Cons_eq: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list"   and ys :: "'a list" assumes "\<not> x < y"   and "\<not> y < x"   and "lexordp xs ys" shows "lexordp (x # xs) (y # ys)"
pconj_ge_one: fixes a :: "real"   and b :: "real" assumes "1 \<le> a + b" shows "a .& b = a + b - 1"
a': shows "( \<Gamma> \<Rightarrow>* \<Delta> \<oplus> F \<nabla> [x].A, n) \<in> derivable  R*"
noWhileL_def: fixes cl :: "('d, 'e, 'f) cmd list" shows "noWhileL cl \<equiv> \<forall>c\<in>set cl. noWhile c"
digit_decode_encode_lt: fixes a :: "nat"   and ds :: "nat list" assumes "a < prod_list ds" shows "digit_decode ds (digit_encode ds a) = a"
renaming: fixes xvec :: "name list"   and Tvec :: "'b list"   and p :: "(name \<times> name) list"   and T :: "'a" assumes "length xvec = length Tvec"   and "set p \<subseteq> set xvec \<times> set (p \<bullet> xvec)"   and "distinctPerm p"   and "(p \<bullet> xvec) \<sharp>* T" shows "T[xvec::=Tvec] = p \<bullet> T[p \<bullet> xvec::=Tvec]"
eq_set_tree_empty: fixes t :: "('c \<times> 'd) tree" shows "(Tree2.set_tree t = {}) = (t = \<langle>\<rangle>)"
L: shows "(map (inv p) (filter (\<lambda>x. x \<in> range p) (xs @ y # ys)), inv p ` R) \<in> failures P"
0: shows "prv (imp (cnj \<phi> (cnj \<phi>1 (lcnj (\<phi>1s @ \<phi>2s)))) (cnj \<phi>1 (cnj \<phi> (lcnj (\<phi>1s @ \<phi>2s)))))"
subs_2: shows "{Inl x |x. x \<in> SIGT t2} \<subseteq> {Inl x |x. x \<in> SIGT (Times t1 t2)}"
target: fixes T1 :: "'a"   and T2 :: "'a"   and TRel :: "('a \<times> 'a) set"   and Enc :: "'b \<Rightarrow> 'a"   and SRel :: "('b \<times> 'b) set" assumes "(T1, T2) \<in> TRel" shows "(TargetTerm T1, TargetTerm T2) \<in> encoding.indRelSTEQ Enc SRel TRel"
et_pair: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e1 :: "expr"   and t1 :: "pdf_type"   and e2 :: "expr"   and t2 :: "pdf_type" assumes "\<Gamma> \<turnstile> e1 : t1"   and "\<Gamma> \<turnstile> e2 : t2" shows "\<Gamma> \<turnstile> <e1 ,  e2> : PRODUCT t1 t2"
cons: fixes y :: "'a"   and xs :: "'a list"   and x :: "'a" assumes "P (y # xs)" shows "P (x # y # xs)"
conc_Diff_if_Nil1: fixes A :: "'b list set"   and B :: "'b list set" assumes "[] \<in> A" shows "A @@ B = (A - {[]}) @@ B \<union> B"
Its_0: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and c :: "com" assumes "\<not> bval b s" shows "Its b c s 0"
MinkowskiBetweenness_def: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" shows "MinkowskiBetweenness \<E> \<P> betw \<equiv> MinkowskiPrimitive \<E> \<P> \<and> MinkowskiBetweenness_axioms \<P> betw"
canonical_saturated_3: fixes X :: "'b set"   and v :: "'b \<Rightarrow> nat"   and n :: "nat"   and not_in_X :: "'b"   and r :: "'c"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and c1 :: "'b"   and c2 :: "'b" assumes "Beta_Regions' X v n not_in_X"   and "Le r \<le> M (v c1) (v c2)"   and "Le (- r) \<le> M (v c2) (v c1)"   and "cycle_free M n"   and "canonical M n"   and "v c1 \<le> n"   and "v c2 \<le> n"   and "v c1 \<noteq> v c2"   and "\<forall>c. v c \<le> n \<longrightarrow> 0 < v c" obtains u :: "'b \<Rightarrow> 'c" where "u \<in> [M]\<^bsub>v,n\<^esub>" and "u c1 - u c2 = r"
eventually_at_left_real: fixes b :: "real"   and a :: "real" assumes "b < a" shows "\<forall>\<^sub>F x in at_left a. x \<in> {b<..<a}"
binop_dfa_steps: fixes bs :: "bool list list"   and f :: "bool \<Rightarrow> bool \<Rightarrow> bool" assumes "list_all (is_alph n) bs" shows "snd (binop_dfa f A B) ! dfa_steps (binop_dfa f A B) 0 bs = f (snd A ! dfa_steps A 0 bs) (snd B ! dfa_steps B 0 bs)"
S2: shows "c_pair 1 0 = Suc (Suc 0)"
RedThrowNull: fixes s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" shows "P \<turnstile> \<langle>throw null,s,b\<rangle> \<rightarrow> \<langle>THROW NullPointer,s,b\<rangle>"
decompose_com_small_step: fixes c' :: "('g, 'h, 'i, 'j) com"   and fctxt :: "('g, 'h, 'i, 'j) com \<Rightarrow> ('g, 'h, 'i, 'j) com list"   and cs :: "('g, 'h, 'i, 'j) com list"   and s :: "'h option \<times> 'j"   and \<alpha> :: "('g, 'i) seq_label"   and s' :: "('g, 'h, 'i, 'j) com list \<times> 'h option \<times> 'j"   and ictxt :: "('g, 'h, 'i, 'j) com \<Rightarrow> ('g, 'h, 'i, 'j) com"   and c :: "('g, 'h, 'i, 'j) com" assumes "(c' # fctxt c' @ cs, s) \<rightarrow>\<^bsub>\<alpha>\<^esub> s'"   and "(c', ictxt, fctxt) \<in> decompose_com c" shows "(c # cs, s) \<rightarrow>\<^bsub>\<alpha>\<^esub> s'"
simps_3: fixes x :: "int"   and y :: "int"   and xs :: "int list" shows "take_decr (x # y # xs) = (if y \<le> x then x # take_decr (y # xs) else [x])"
LeadsETo_Trans: fixes F :: "'d program"   and A :: "'d set"   and CC :: "'d set set"   and B :: "'d set"   and C :: "'d set" assumes "F \<in> A LeadsTo[CC] B"   and "F \<in> B LeadsTo[CC] C" shows "F \<in> A LeadsTo[CC] C"
inverse_unique: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and u :: "'c"   and v' :: "'c"   and v :: "'c" assumes "monoid M composition unit"   and "composition u v' = unit"   and "composition v u = unit"   and "u \<in> M"   and "v \<in> M"   and "v' \<in> M" shows "v = v'"
fv_subterms_set_subst: fixes \<theta> :: "'c \<Rightarrow> ('b, 'a) Term.term"   and M :: "('b, 'c) Term.term set" shows "fv\<^sub>s\<^sub>e\<^sub>t (subterms\<^sub>s\<^sub>e\<^sub>t M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>) = fv\<^sub>s\<^sub>e\<^sub>t (subterms\<^sub>s\<^sub>e\<^sub>t (M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>))"
elims_2: fixes x :: "nat"   and xa :: "nat"   and xb :: "cell list \<times> cell list" assumes "wadjust_loop_right_move x xa xb" obtains m :: "nat" and rs :: "nat" and l :: "cell list" and r :: "cell list" where "x = m" and "xa = rs" and "xb = (l, r)" and "\<exists>ml mr nl nr rn. l = Bk \<up> nl @ Oc # Oc \<up> ml @ Bk # Oc \<up> Suc m \<and> r = Bk \<up> nr @ Oc \<up> mr @ Bk \<up> rn \<and> ml + mr = Suc (Suc rs) \<and> 0 < mr \<and> 0 < nl + nr"
transitive: fixes P :: "ccs"   and Q :: "ccs"   and R :: "ccs" assumes "P \<cong> Q"   and "Q \<cong> R" shows "P \<cong> R"
intros_3: fixes u :: "'e \<Rightarrow> 'd"   and r :: "'e"   and d :: "'d" assumes "u r < d" shows "dbm_entry_val u (Some r) None (Lt d)"
simps_1: fixes y :: "arctic" shows "MinInfty + y = y"
rotate_is_Nil_conv: fixes n :: "nat"   and xs :: "'a list" shows "(rotate n xs = []) = (xs = [])"
att_elem_seq: fixes x1 :: "'a attree"   and x :: "'a" assumes "\<turnstile>x1"   and "x \<in> fst (attack x1)" shows "\<exists>y. y \<in> snd (attack x1) \<and> x \<rightarrow>\<^sub>i* y"
zippers_view_same_hash: fixes z :: "(((view_metadata, view_metadata) blindable\<^sub>m \<times> (view_data, view_data) blindable\<^sub>m) \<times> view\<^sub>m list \<times> view\<^sub>m list) list \<times> view\<^sub>m"   and p :: "((view_metadata \<times> view_data) \<times> view list \<times> view list) list"   and t :: "view" assumes "z \<in> set (zippers_view (p, t))" shows "hash_view (view_of_zipper\<^sub>m z) = hash_view (view_of_zipper\<^sub>m (embed_view_path p, embed_view t))"
add_increasing2: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and c :: "'d"   and b :: "'d"   and a :: "'d" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq zero c"   and "less_eq b a" shows "less_eq b (plus a c)"
Diagonal_to_Smith_row_i_PQ_JNF_def: fixes i :: "nat"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and P :: "'b mat"   and A :: "'b mat"   and Q :: "'b mat" shows "Diagonal_to_Smith_row_i_PQ_JNF i bezout (P, A, Q) = diagonal_to_Smith_i_PQ_JNF [i + 1..<min (dim_row A) (dim_col A)] i bezout (P, A, Q)"
mono2mono_eadd: fixes orda :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and t :: "'a \<Rightarrow> enat"   and t' :: "'a \<Rightarrow> enat" assumes "monotone orda (\<le>) t"   and "monotone orda (\<le>) t'" shows "monotone orda (\<le>) (\<lambda>x. t x + t' x)"
sat_\<Phi>: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "I.wtE \<xi>" shows "I.satPB \<xi> \<Phi>"
rev_upt_Suc: fixes i :: "nat"   and j :: "nat" shows "[Suc i>..j] = (if j \<le> i then i # [i>..j] else [])"
supp_plus_perm_eq: fixes p :: "perm"   and q :: "perm" assumes "supp p \<inter> supp q = {}" shows "supp (p + q) = supp p \<union> supp q"
prime_field: fixes q :: "nat"   and a :: "nat" assumes "prime q"   and "a < q"   and "a \<noteq> 0" shows "coprime a q"
intro: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c" assumes "ChamberComplex X"   and "ChamberComplex Y"   and "ChamberComplexMorphism_axioms X Y f" shows "ChamberComplexMorphism X Y f"
n_comm_var: fixes x :: "'a"   and y :: "'a" shows "!(!x \<cdot> !y) = !(!y \<cdot> !x)"
HMSet_plus: fixes A :: "hmultiset multiset"   and B :: "hmultiset multiset" shows "HMSet (A + B) = HMSet A + HMSet B"
succ_def: fixes G :: "('e, 'd) graph"   and v :: "'e" shows "succ G v \<equiv> {(w, v'). (v, w, v') \<in> edges G}"
round_def: fixes x :: "'a" shows "round x = \<lfloor>x + (1::'a) / (2::'a)\<rfloor>"
s: shows "is_arg_min norm (\<lambda>t. t \<in> M) s"
row_greater_greatest_is_zero: fixes k :: "nat"   and A :: "(('b, 'c) vec, 'a) vec"   and i :: "'a" assumes "(GREATEST m. \<not> is_zero_row_upt_k m k A) < i" shows "is_zero_row_upt_k i k A"
avl_dist2: fixes n :: "'b"   and l :: "'b tree"   and r :: "'b tree"   and h :: "nat"   and x :: "'b" assumes "avl (MKT n l r h)"   and "is_ord (MKT n l r h)"   and "x \<in> set_of l \<or> x \<in> set_of r" shows "x \<noteq> n"
comm_semiring_0_cancel_ow_def: fixes U :: "'a set"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "comm_semiring_0_cancel_ow U plus minus zero times \<equiv> comm_semiring_ow U plus times \<and> cancel_comm_monoid_add_ow U plus minus zero"
word_rcat_def: shows "word_rcat = word_of_int \<circ> horner_sum uint (2 ^ LENGTH('b)) \<circ> rev"
APP_def: fixes f :: "'d \<Rightarrow> 'c"   and a :: "'d" shows "f $ a \<equiv> f a"
bounded_iff: fixes a :: "nat"   and b :: "nat"   and c :: "nat" shows "(a dvd gcd b c) = (a dvd b \<and> a dvd c)"
PP_def: shows "PP \<equiv> (B\<^sub>1 ===> B\<^sub>1 ===> B\<^sub>1) ===> B\<^sub>1 ===> (B\<^sub>1 ===> B\<^sub>1 ===> B\<^sub>1) ===> (B\<^sub>1 ===> B\<^sub>1) ===> ((=) ===> B\<^sub>1 ===> B\<^sub>1) ===> (B\<^sub>2 ===> B\<^sub>2 ===> B\<^sub>2) ===> B\<^sub>2 ===> (B\<^sub>2 ===> B\<^sub>2 ===> B\<^sub>2) ===> (B\<^sub>2 ===> B\<^sub>2) ===> ((=) ===> B\<^sub>2 ===> B\<^sub>2) ===> (B\<^sub>1 ===> B\<^sub>2) ===> (=)"
arr_inv: fixes f :: "'s" assumes "local.iso f" shows "arr (local.inv f)"
assms: shows "(red F)\<^sup>*\<^sup>* (p - q) 0"
c: shows "c \<in> inside (path_image (p +++ linepath (a - complex_of_real e) (a + complex_of_real e)))"
WT: fixes x :: "'call"   and s :: "'s" assumes "x \<in> outs_\<I> \<I>" shows "\<I>' \<turnstile>g callee s x \<surd>"
simps_2: fixes carrier :: "'c set"   and mult :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and more :: "'d" shows "\<one>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, \<dots> = more\<rparr>\<^esub> = one"
preserves_reflects_arr: fixes f :: "'a arrow_of_spans_data" shows "V.arr (trg f) = V.arr f"
rec_sigma2_def: fixes f :: "recf" shows "rec_sigma2 f = PR (CN f [CN Z [recf.Id 2 0], recf.Id 2 0, recf.Id 2 1]) (CN rec_add [recf.Id 4 1, CN f [CN S [recf.Id 4 0], recf.Id 4 2, recf.Id 4 3]])"
wt_ge_num_args: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> nat"   and s :: "('b, 'a) tm" assumes "kbo_std ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym"   and "arity.wary arity_sym arity_var s" shows "num_args s \<le> kbo_std.wt ground_heads_var \<delta> arity_sym wt_sym s"
simps: fixes ars :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and a1 :: "'a"   and a2 :: "(bool \<times> 'b \<times> 'a) list" shows "convp ars a1 a2 = ((\<exists>a. a1 = a \<and> a2 = []) \<or> (\<exists>a \<alpha> b ss. a1 = a \<and> a2 = (True, \<alpha>, b) # ss \<and> ars a \<alpha> b \<and> convp ars b ss) \<or> (\<exists>b \<alpha> a ss. a1 = a \<and> a2 = (False, \<alpha>, b) # ss \<and> ars b \<alpha> a \<and> convp ars b ss))"
smallo_imp_asymp_equiv: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "(\<lambda>x. f x - g x) \<in> o[F](g)" shows "f \<sim>[F] g"
basis_image_linear: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'b) vec"   and X :: "(('a, 'b) vec, 'c) vec" assumes "invertible_lf (*s) (*s) f"   and "is_basis (set_of_vector X)" shows "is_basis (f ` set_of_vector X)"
Un_left_absorb: fixes A :: "'b set"   and B :: "'b set" shows "(A::'b set) \<union> (A \<union> (B::'b set)) = A \<union> B"
\<tau>move2_sync2D: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and V :: "nat"   and o' :: "'addr expr1"   and e :: "'addr expr1"   and pc :: "nat"   and xcp :: "'addr option" assumes "\<tau>move2 P h stk (sync\<^bsub>V\<^esub> (o') e) (Suc (Suc (Suc (length (compE2 o') + pc)))) xcp"   and "pc < length (compE2 e)" shows "\<tau>move2 P h stk e pc xcp"
acc_modi_le_Package: fixes a :: "acc_modi" assumes "a \<le> Package" shows "a = Private \<or> a = Package"
ltl_to_generalized_rabin_correct: fixes \<Sigma> :: "'b set set"   and w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltl" assumes "ltl_to_rabin_af \<Sigma> w" shows "w \<Turnstile> \<phi> = accept\<^sub>G\<^sub>R (ltl_to_rabin_base_def.ltl_to_generalized_rabin \<up>af \<up>af\<^sub>G Abs Abs M_fin \<Sigma> \<phi>) w"
convex_rel_interior: fixes S :: "'b set" assumes "convex S" shows "convex (rel_interior S)"
wf_lexn: fixes r :: "('g \<times> 'g) set"   and n :: "nat" assumes "wf r" shows "wf (lexn r n)"
minusinf_inf: fixes p :: "fm"   and a :: "real"   and bs :: "real list" assumes "iszlfm p (a # bs)" shows "\<exists>z. \<forall>x<z. Ifm (real_of_int x # bs) (minusinf p) = Ifm (real_of_int x # bs) p"
eprod_rule: fixes p :: "'c \<times> 'd"   and S :: "'c \<Rightarrow> 'd \<Rightarrow> ('e + 'f) nres"   and \<Phi> :: "'e \<Rightarrow> bool"   and \<Psi> :: "'f \<Rightarrow> bool" assumes "\<And>a b. p = (a, b) \<Longrightarrow> S a b \<le> ESPEC \<Phi> \<Psi>" shows "(case p of (a, b) \<Rightarrow> S a b) \<le> ESPEC \<Phi> \<Psi>"
vsubset_vunion_leftI: fixes A :: "V"   and C :: "V"   and B :: "V" assumes "A \<subseteq>\<^sub>\<circ> C"   and "B \<subseteq>\<^sub>\<circ> C" shows "A \<union>\<^sub>\<circ> B \<subseteq>\<^sub>\<circ> C"
map_poly_hom_add: fixes p :: "'a poly"   and q :: "'a poly" shows "map_poly hom (p + q) = map_poly hom p + map_poly hom q"
hyps_1: shows "s\<^sub>2 = s\<^sub>1(r \<mapsto> (\<sigma>, \<tau>, \<E> [VE (Rid r')]), r' \<mapsto> (\<sigma>;;\<tau>, \<epsilon>, e))"
VDM_validn_def: fixes n :: "nat"   and c :: "OBJ"   and A :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" shows "( \<Turnstile>\<^sub>n c : A ) = (\<forall>m\<le>n. \<forall>s t.  s , c \<rightarrow>\<^sub>m  t  \<longrightarrow> A s t)"
bij: shows "bij_betw (map_prod post post) (set (map snd ws @ ps)) N"
Am_def: fixes t :: "'b tree" shows "S34.Am t = real (T_splay_max t) + S34.\<Phi> (splay_max t) - S34.\<Phi> t"
fps_inv: fixes a :: "'a fps" assumes "a $ 0 = (0::'a)"   and "a $ 1 \<noteq> (0::'a)" shows "fps_inv a oo a = fps_X"
rep_props_4: fixes r :: "'a" assumes "ide r" shows "isomorphic_rep r (tab\<^sub>0 r) (tab\<^sub>1 r)"
extreme_boundI: fixes X :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and s :: "'c" assumes "\<And>b. \<lbrakk>bound X less_eq b; b \<in> A\<rbrakk> \<Longrightarrow> less_eq s b"   and "\<And>x. x \<in> X \<Longrightarrow> less_eq x s"   and "s \<in> A" shows "extreme_bound A less_eq X s"
nonneg: fixes x :: "nat" shows "(0::'a) \<le> f x"
abs_eq: fixes x :: "tag" shows "Tag x = Ab (cTag x)"
subresultant'_mat_def: fixes J :: "nat"   and l :: "nat"   and F :: "'b poly"   and G :: "'b poly" shows "subresultant'_mat J l F G = (let \<gamma> = degree G; \<phi> = degree F; f = coeff_int F; g = coeff_int G; n = \<phi> - J + (\<gamma> - J) in mat n n (\<lambda>(i, j). if j < \<gamma> - J then if i = n - 1 then f (int l - int (\<gamma> - J - 1) + int j) else f (int \<phi> - int i + int j) else let jj = j - (\<gamma> - J) in if i = n - 1 then g (int l - int (\<phi> - J - 1) + int jj) else g (int \<gamma> - int i + int jj)))"