hcju/isabelleps huggingface dataset:
coinduct: fixes G :: "('c, 'd) ParityGame_scheme"   and X :: "Player \<Rightarrow> 'c llist \<Rightarrow> ('c \<Rightarrow> 'c) \<Rightarrow> bool"   and x1 :: "Player"   and x2 :: "'c llist"   and x3 :: "'c \<Rightarrow> 'c" assumes "ParityGame G"   and "X x1 x2 x3"   and "\<And>x1 x2 x3. X x1 x2 x3 \<Longrightarrow> (\<exists>p \<sigma>. x1 = p \<and> x2 = LNil \<and> x3 = \<sigma>) \<or> (\<exists>p v \<sigma>. x1 = p \<and> x2 = LCons v LNil \<and> x3 = \<sigma>) \<or> (\<exists>v p w \<sigma> Ps. x1 = p \<and> x2 = LCons v (LCons w Ps) \<and> x3 = \<sigma> \<and> v \<in> (if p = Even then V0\<^bsub>G\<^esub> else V\<^bsub>G\<^esub> - V0\<^bsub>G\<^esub>) \<and> w = \<sigma> v \<and> (X p (LCons w Ps) \<sigma> \<or> ParityGame.path_conforms_with_strategy G p (LCons w Ps) \<sigma>)) \<or> (\<exists>v p Ps \<sigma>. x1 = p \<and> x2 = LCons v Ps \<and> x3 = \<sigma> \<and> v \<notin> (if p = Even then V0\<^bsub>G\<^esub> else V\<^bsub>G\<^esub> - V0\<^bsub>G\<^esub>) \<and> (X p Ps \<sigma> \<or> ParityGame.path_conforms_with_strategy G p Ps \<sigma>))" shows "ParityGame.path_conforms_with_strategy G x1 x2 x3"
class_rec_def: shows "class_rec \<equiv> wfrec (subcls1\<inverse>) (\<lambda>rec C f. case class C of Some m \<Rightarrow> (if C = Object then Map.empty else rec (super m) f) ++ map_of (f m))"
of_nat_eq_iff: fixes m :: "nat"   and n :: "nat" shows "(of_nat m = of_nat n) = (m = n)"
sat_precond_exec_as_proj_eq_proj_exec: fixes s :: "('d, 'e) fmap"   and as :: "(('d, 'e) fmap \<times> ('d, 'e) fmap) list"   and vs :: "'d set" assumes "sat_precond_as s as" shows "exec_plan (fmrestrict_set vs s) (as_proj as vs) = fmrestrict_set vs (exec_plan s as)"
seqI': fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a"   and g :: "'a \<times> 'a"   and c :: "'a \<times> 'a" assumes "CC.in_hom f a b"   and "CC.in_hom g b c" shows "CC.seq g f"
Act\<^sub>\<alpha>_eq_iff: fixes f1 :: "'e"   and \<alpha>1 :: "'d"   and t1 :: "('b, 'c, 'd, 'e) Tree\<^sub>\<alpha>"   and f2 :: "'e"   and \<alpha>2 :: "'d"   and t2 :: "('b, 'c, 'd, 'e) Tree\<^sub>\<alpha>" shows "(Act\<^sub>\<alpha> f1 \<alpha>1 t1 = Act\<^sub>\<alpha> f2 \<alpha>2 t2) = (tAct f1 \<alpha>1 (rep_Tree\<^sub>\<alpha> t1) =\<^sub>\<alpha> tAct f2 \<alpha>2 (rep_Tree\<^sub>\<alpha> t2))"
least: fixes f :: "'f" shows "least f = (w (f, 0) = w0 \<and> (\<forall>g. w (g, 0) = w0 \<longrightarrow> pr_weak (g, 0) (f, 0)))"
boundOutputFreshSimps_4: fixes x :: "name"   and xvec :: "name list"   and B :: "('d, 'e, 'f) boundOutput" assumes "x \<sharp> xvec" shows "x \<sharp> \<lparr>\<nu>*xvec\<rparr>B = x \<sharp> B"
\<Omega>_1: shows "\<Omega> \<inter> space M \<in> sets M"
comp_ide_self: fixes a :: "'a \<times> 'a" assumes "H.VV.ide a" shows "H.VV.comp a a = a"
simps: fixes e :: "nat"   and n :: "nat" shows "local.encode e n = (if n = 0 then 0 else of_nat (n mod base) * \<omega>^ (local.encode 0 e) + local.encode (e + 1) (n div base))"
mem_case_prodI: fixes z :: "'b"   and c :: "'c \<Rightarrow> 'd \<Rightarrow> 'b set"   and a :: "'c"   and b :: "'d" assumes "z \<in> c a b" shows "z \<in> (case (a, b) of (d, e) \<Rightarrow> c d e)"
assms: shows "(+) (- a) ` V \<subseteq> S"
BOresChainEqvt: fixes perm :: "(name \<times> name) list"   and xvec :: "name list"   and B :: "('d, 'e, 'f) boundOutput" shows "perm \<bullet> \<lparr>\<nu>*xvec\<rparr>B = \<lparr>\<nu>*perm \<bullet> xvec\<rparr>perm \<bullet> B"
word_of_int_eq_iff: fixes k :: "int"   and l :: "int" shows "(word_of_int k = word_of_int l) = (take_bit LENGTH('b) k = take_bit LENGTH('b) l)"
cases: fixes a :: "'b list"   and A :: "'b set"   and n :: "nat" assumes "a \<in> nlists A n" obtains xs :: "'b list" where "a = xs" and "set xs \<subseteq> A" and "length xs = n"
shiftr_mask: fixes m :: "nat" shows "mask m >> m = 0"
cSup_subset_mono: fixes A :: "'d multiset set"   and B :: "'d multiset set" assumes "A \<noteq> {}"   and "subset_mset.bdd_above B"   and "A \<subseteq> B" shows "Sup A \<subseteq># Sup B"
meet_join_distrib2: fixes L :: "('c, 'd) boolean_algebra_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "boolean_algebra L"   and "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "(y \<squnion>\<^bsub>L\<^esub> z) \<sqinter>\<^bsub>L\<^esub> x = y \<sqinter>\<^bsub>L\<^esub> x \<squnion>\<^bsub>L\<^esub> z \<sqinter>\<^bsub>L\<^esub> x"
intro: fixes invar :: "'c \<Rightarrow> bool"   and iteratei :: "'c \<Rightarrow> ('d \<Rightarrow> bool) \<Rightarrow> ('e \<times> 'f \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and \<alpha> :: "'c \<Rightarrow> 'e \<Rightarrow> 'f option" assumes "\<And>m. invar m \<Longrightarrow> map_iterator (iteratei m) (\<alpha> m)" shows "map_iteratei_axioms \<alpha> invar iteratei"
terminals_def: fixes G :: "diagram" shows "terminals G = ffilter (\<lambda>v. \<forall>e\<in>set G^E. v |\<notin>| fst3 e) G^V"
splitFace_distinct2: fixes g :: "graph"   and ram1 :: "nat"   and ram2 :: "nat"   and oldF :: "face"   and newVertexList :: "nat list" assumes "pre_splitFace g ram1 ram2 oldF newVertexList" shows "distinct (vertices (fst (splitFace g ram1 ram2 oldF newVertexList)))"
prime_factorization_nat'_def: fixes n :: "nat" shows "prime_factorization_nat' n = (let ps = prime_factorization_nat n in map (\<lambda>p. (p, length (filter ((=) p) ps) - 1)) (remdups_adj (sort ps)))"
domain_Epigraph_fst: fixes f :: "'b \<Rightarrow> ereal" shows "domain f = fst ` Epigraph UNIV f"
kruskal_correct: fixes L :: "(nat \<times> int \<times> nat) list" assumes "fromlist L" shows "<emp> kruskal (return L) (\<lambda>(u, w, v). return (u, v)) () <\<lambda>r. \<up> (distinct r \<and> set r \<subseteq> set L \<and> uGraph.minSpanningForest (uGraph_from_list_\<alpha>_edges L) (uGraph_from_list_\<alpha>_weight L) (set (map \<alpha> r)))>\<^sub>t"
sum_mult_sum_if_inj: fixes f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and A :: "'a set"   and B :: "'b set" assumes "inj_on (\<lambda>(a, b). f a * g b) (A \<times> B)" shows "sum f A * sum g B = sum id {f a * g b |a b. a \<in> A \<and> b \<in> B}"
ordinal_times_oSuc: fixes x :: "ordinal"   and y :: "ordinal" shows "x * oSuc y = x * y + x"
seqSubsTermAppend: fixes P :: "('a, 'b, 'c) psi"   and \<sigma> :: "(name list \<times> 'a list) list"   and \<sigma>' :: "(name list \<times> 'a list) list" shows "P[<(\<sigma> @ \<sigma>')>] = P[<\<sigma>>][<\<sigma>'>]"
rs_s_ts_2: shows "\<exists>B\<in>set Cs. \<forall>(f, g)\<in>fundfoldpairs. s = Abs_induced_automorph f g \<longrightarrow> B \<in> g \<turnstile> \<C>"
mset_subset_diff_self: fixes c :: "'b"   and B :: "'b multiset" assumes "c \<in># B" shows "remove1_mset c B \<subset># B"
ind_hyp: fixes y :: "'a list"   and sl :: "'a list" assumes "length y < length xs" shows "quicksort_by_rel R sl y = quicksort_by_rel R [] y @ sl"
6: shows "w\<^sup>T\<^sup>\<star> * d\<^sup>T * top \<le> - ((w \<sqinter> - d)\<^sup>T\<^sup>\<star> * e\<^sup>T * top)"
induct: fixes P :: "'a \<Rightarrow> 'node \<Rightarrow> 'var \<Rightarrow> bool"   and a0 :: "'a"   and a1 :: "'node"   and a2 :: "'var" assumes "\<And>g n v. (\<And>x21 x22. \<lbrakk>\<not> n \<notin> set (\<alpha>n2 g); v \<notin> defs2 g n; i.predecessors g n = x21 # x22; x22 = []\<rbrakk> \<Longrightarrow> P g x21 v) \<Longrightarrow> P g n v" shows "P a0 a1 a2"
Hermite_reduce_above_preserves: fixes n :: "nat"   and a :: "'a"   and A :: "(('b, 'c) vec, 'a) vec"   and i :: "'a"   and j :: "'c"   and res :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'c" assumes "n \<le> mod_type_class.to_nat a" shows "Hermite_reduce_above A n i j res $ a $ b = A $ a $ b"
methd_rT_is_type: fixes G :: "prog"   and C :: "qtname"   and sig :: "sig"   and m :: "qtname \<times> methd"   and y :: "class" assumes "wf_prog G"   and "methd G C sig = Some m"   and "class G C = Some y" shows "is_type G (resTy m)"
set_intros_1: fixes x21 :: "'l"   and x22 :: "'l list" shows "x21 \<in> set (x21 # x22)"
wW: fixes a :: "'a mod_ring poly"   and b :: "'a mod_ring poly"   and u :: "'b mod_ring poly"   and v :: "'c mod_ring poly"   and w :: "'c mod_ring poly"   and q_ty :: "'c itself" assumes "Knuth_ex_4_6_2_22_main a b u v w" shows "w = #(#w + Polynomial.smult (of_nat (Knuth_ex_4_6_2_22_base.q q_ty)) (#Knuth_ex_4_6_2_22_main.w' q_ty a b u v w))"
relcompp_apply: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and S :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'c"   and c :: "'e" shows "(R OO S) a c = (\<exists>b. R a b \<and> S b c)"
take_strict: fixes n :: "nat" shows "Box_take n\<cdot>\<bottom> = \<bottom>"
transfer: shows "rel_fun cr_gen_cfg_wf (=) gen_inEdges gen_wf_inEdges"
catalan_Suc: fixes n :: "nat" shows "catalan (Suc n) = (\<Sum>i\<le>n. catalan i * catalan (n - i))"
HK_4: shows "H <#>\<^bsub>homology_group p (subtopology X S)\<^esub> K = carrier (homology_group p (subtopology X S))"
not_in_apply_ncrit: fixes ic :: "bool"   and p :: "('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f"   and q :: "('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f"   and crit :: "nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> bool \<Rightarrow> (bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> bool"   and data :: "nat \<times> 'g"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and xs :: "(bool \<times> bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and q_in_bs :: "bool"   and ys :: "(bool \<times> bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" assumes "(ic, p, q) \<notin> set (apply_ncrit crit data gs bs hs (xs @ (ic, q_in_bs, p, q) # ys))" shows "crit data gs bs hs q_in_bs (rev (apply_ncrit crit data gs bs hs xs)) p q"
vifunionE: fixes a :: "V"   and I :: "V"   and f :: "V \<Rightarrow> V" assumes "a \<in>\<^sub>\<circ> \<Union>\<^sub>\<circ> (VLambda I f `\<^sub>\<circ> I)" obtains i :: "V" where "i \<in>\<^sub>\<circ> I" and "a \<in>\<^sub>\<circ> f i"
simps_2: fixes a :: "'i \<times> 'a atom"   and as :: "('i \<times> 'a atom) list"   and s :: "'s" shows "assert_all_s (a # as) s = (case assert_s a s of Inl x \<Rightarrow> Inl x | Inr x \<Rightarrow> assert_all_s as x)"
sums_subst: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b"   and z :: "'b" assumes "f = g"   and "g sums z" shows "f sums z"
list_order_trans: fixes op_list :: "('a \<times> 'a option) list"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "insert_ops op_list"   and "list_order op_list x y"   and "list_order op_list y z" shows "list_order op_list x z"
tendsto_at_iff_sequentially: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'b"   and x :: "'a"   and s :: "'a set" shows "(f \<longlongrightarrow> a) (at x within s) = (\<forall>X. (\<forall>i. X i \<in> s - {x}) \<longrightarrow> X \<longlonglongrightarrow> x \<longrightarrow> (f \<circ> X) \<longlonglongrightarrow> a)"
a1: shows "low_equal s1 s2 \<and> t1 = snd (fst (set_annul True (snd (fst (write_cpu (cpu_reg_val nPC s2 + 4) nPC (snd (fst (write_cpu (cpu_reg_val nPC s2) PC s1)))))))) \<and> t2 = snd (fst (set_annul True (snd (fst (write_cpu (cpu_reg_val nPC s2 + 4) nPC (snd (fst (write_cpu (cpu_reg_val nPC s2) PC s2))))))))"
reals_Archimedean2: fixes x :: "'c" shows "\<exists>n. x < of_nat n"
enumerate_Suc_eq: fixes n :: "nat"   and xs :: "'a list" shows "enumerate (Suc n) xs = map (apfst Suc) (enumerate n xs)"
wf_set_subset: fixes I :: "nat set"   and V :: "nat set"   and n :: "nat" assumes "I \<subseteq> V"   and "1 \<le> card I"   and "wf_set n V" shows "wf_set n I"
sv_lv_rel_inverse: shows "single_valued (lv_rel\<inverse>)"
chars_of_token_def: fixes t :: "symbol \<times> character list" shows "chars_of_token t = snd t"
filter_insort: fixes f :: "'c \<Rightarrow> 'a"   and xs :: "'c list"   and P :: "'c \<Rightarrow> bool"   and x :: "'c" assumes "sorted (map f xs)"   and "P x" shows "filter P (insort_key f x xs) = insort_key f x (filter P xs)"
subst_support_if_mgt_subst_idem: fixes \<delta> :: "'d \<Rightarrow> ('e, 'f) Term.term"   and \<theta> :: "'d \<Rightarrow> ('e, 'd) Term.term" assumes "\<theta> \<preceq>\<^sub>\<circ> \<delta>"   and "subst_idem \<theta>" shows "\<theta> supports \<delta>"
next_subseqs1_set_as_next_subseqs2_set: fixes i :: "nat"   and v :: "'b"   and prevs :: "(nat \<times> 'b) list" shows "next_subseqs1_set ((i, v) # prevs) = next_subseqs1_set prevs \<union> next_subseqs2_set v [0..<i]"
trace_parallel_plan_plan_prefix: fixes k :: "nat"   and I :: "'b \<Rightarrow> bool option"   and \<pi> :: "'b strips_operator list list" assumes "k < length (trace_parallel_plan_strips I \<pi>)" shows "trace_parallel_plan_strips I \<pi> ! k = execute_parallel_plan I (take k \<pi>)"
antisym_conv: fixes y :: "'a"   and x :: "'a" assumes "y \<le> x" shows "(x \<le> y) = (x = y)"
fac: shows "\<Inter> {S \<inter> {x. a h \<bullet> x = b h} |h. h \<in> F \<and> C \<subseteq> S \<inter> {x. a h \<bullet> x = b h}} face_of S"
rep_eq: fixes x :: "natural"   and xa :: "natural" shows "nat_of_natural (x + xa) = nat_of_natural x + nat_of_natural xa"
\<psi>_\<phi>: fixes y :: "'b"   and f :: "'a"   and x :: "'a" assumes "B.ide y"   and "\<guillemotleft>f : F y \<rightarrow>\<^sub>A x\<guillemotright>" shows "\<psi> x (\<phi> y f) = f"
FactGroup_subset: fixes g :: "'a"   and g' :: "'a" assumes "g \<in> carrier G"   and "g' \<in> carrier G"   and "h g = h g'" shows "kernel G H h #> g \<subseteq> kernel G H h #> g'"
cases: fixes x :: "('b \<Rightarrow> exp \<Rightarrow> exp) \<times> 'b\<^sub>\<bottom> \<times> exp" obtains t :: "'b \<Rightarrow> exp \<Rightarrow> exp" and e :: "exp" where "x = (t__, Ibottom, e__)"   | t :: "'b \<Rightarrow> exp \<Rightarrow> exp" and a :: "'b" and e :: "exp" where "x = (t, Iup a, e)"
strict_sorted_h: shows "strict_sorted (sorted_list_of_set {x. \<exists>q\<in>set qs. q \<noteq> 0 \<and> poly q x = 0})"
vec_lambda_inject: fixes x :: "'a \<Rightarrow> 'b"   and y :: "'a \<Rightarrow> 'b" assumes "x \<in> UNIV"   and "y \<in> UNIV" shows "(vec_lambda x = vec_lambda y) = (x = y)"
prems_3: fixes y :: "'a" assumes "y \<in> set (y__ # ys)" shows "is_unit (lift y)"
hd_sort_Min: fixes l :: "'b list" assumes "l \<noteq> []" shows "hd (sort l) = Min (set l)"
UN_mono: fixes A :: "'c set"   and B :: "'c set"   and f :: "'c \<Rightarrow> 'd set"   and g :: "'c \<Rightarrow> 'd set" assumes "A \<subseteq> B"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x" shows "\<Union> (f ` A) \<subseteq> \<Union> (g ` B)"
exhaust_sel: fixes option :: "'a option" obtains "option = None"   | "option = \<lfloor>the option\<rfloor>"
n_add_distr: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "!x + !y \<cdot> !z = (!x + !y) \<cdot> (!x + !z)"
closed_injective_linear_image: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "closed S"   and "linear f"   and "inj f" shows "closed (f ` S)"
closed_sequentially: fixes s :: "'b set"   and f :: "nat \<Rightarrow> 'b"   and l :: "'b" assumes "closed s"   and "\<forall>n. f n \<in> s"   and "f \<longlonglongrightarrow> l" shows "l \<in> s"
A0_eq: shows "A0 = P' * A0 * 1\<^sub>m 1"
of_real_add: fixes x :: "real"   and y :: "real" shows "of_real (x + y) = of_real x + of_real y"
antisym_conv3: fixes y :: "'a"   and x :: "'a" assumes "\<not> y \<sqsubset> x" shows "(\<not> x \<sqsubset> y) = (x = y)"
simps_6: fixes \<alpha> :: "('b, 'c, 'a) hp"   and \<beta> :: "('b, 'c, 'a) hp" shows "BVP (\<alpha> \<union>\<union> \<beta>) = BVP \<alpha> \<union> BVP \<beta>"
nu_witness_def: shows "nu_witness \<equiv> mu + lambda - mu mod lambda"
\<sigma>_1: shows "strategy_attracts p \<sigma> A W"
simps_3: fixes m :: "'node"   and x :: "nat" shows "valid_SDG_node (Formal_out (m, x)) = (\<exists>a Q p f ins outs. valid_edge a \<and> kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f \<and> sourcenode a = m \<and> (p, ins, outs) \<in> set procs \<and> x < length outs)"
assms: shows "\<forall>l\<in>dom f. \<forall>x y. Px x y \<longrightarrow> P x y (f l) l"
x_zero_add_closed: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> (0::'a) + y \<cdot> (0::'a) = (x + y) \<cdot> (0::'a)"
square_freeI: fixes p :: "'a poly" assumes "\<And>q. \<lbrakk>0 < degree q; q \<noteq> 0; q * q dvd p\<rbrakk> \<Longrightarrow> False"   and "p \<noteq> 0" shows "square_free p"
h_plus_OUT_le_IN: fixes x :: "'v"   and i :: "nat" assumes "x \<noteq> source \<Delta>" shows "d_OUT (h_plus i) x \<le> d_IN (h_plus i) x"
inf_of_two_exists: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier (inv_gorder L)"   and "y \<in> carrier (inv_gorder L)" shows "\<exists>s. is_glb (inv_gorder L) s {x, y}"
complex_root_unity_eq: fixes n :: "nat"   and j :: "nat"   and k :: "nat" assumes "1 \<le> n" shows "(exp (2 * complex_of_real pi * \<i> * of_nat j / of_nat n) = exp (2 * complex_of_real pi * \<i> * of_nat k / of_nat n)) = (j mod n = k mod n)"
compl_inf: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and bot :: "'e"   and top :: "'e"   and x :: "'e"   and y :: "'e" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "uminus (inf x y) = sup (uminus x) (uminus y)"
sxalloc_type_sound: fixes G :: "prog"   and s1 :: "state"   and s2 :: "state" assumes "G\<turnstile>s1 \<midarrow>sxalloc\<rightarrow> s2"   and "wf_prog G" shows "case fst s1 of None \<Rightarrow> s2 = s1 | Some (Xcpt x) \<Rightarrow> \<exists>a. fst s2 = Some (Xcpt (Loc a)) \<and> (\<forall>L. s1\<Colon>\<preceq>(G, L) \<longrightarrow> s2\<Colon>\<preceq>(G, L)) | Some _ \<Rightarrow> s2 = s1"
inject_1: fixes x3 :: "'d"   and x2 :: "'d"   and x1 :: "('d, 'e, 'f) psi"   and y3 :: "'d"   and y2 :: "'d"   and y1 :: "('d, 'e, 'f) psi" shows "(RIn x3 x2 x1 = RIn y3 y2 y1) = (x3 = y3 \<and> x2 = y2 \<and> x1 = y1)"
less_le: 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" shows "less x y = (less_eq x y \<and> x \<noteq> y)"
inject: fixes a :: "'a"   and b :: "'b"   and a' :: "'a"   and b' :: "'b" shows "((a, b) = (a', b')) = (a = a' \<and> b = b')"
fMin_in: fixes A :: "'a fset" assumes "A \<noteq> {||}" shows "fMin A |\<in>| A"
hom_induced: shows "((\<forall>p X S Y T f c. \<not> (continuous_map X Y f \<and> f ` (topspace X \<inter> S) \<subseteq> T \<and> c \<in> carrier (relative_homology_group p X S)) \<longrightarrow> hom_induced p X S Y T f c = \<one>\<^bsub>relative_homology_group p Y T\<^esub>) \<and> (\<forall>p X S Y T f. hom_induced p X S Y T f \<in> hom (relative_homology_group p X S) (relative_homology_group p Y T)) \<and> (\<forall>p X S Y T f c. continuous_map X Y f \<and> f ` (topspace X \<inter> S) \<subseteq> T \<and> singular_relcycle p X S c \<longrightarrow> hom_induced (int p) X S Y T f (homologous_rel_set p X S c) = homologous_rel_set p Y T (chain_map p f c)) \<and> (\<forall>p X S Y T. hom_induced p X S Y T = hom_induced p X (topspace X \<inter> S) Y (topspace Y \<inter> T))) \<and> (\<forall>p X S Y f T c. hom_induced p X S Y T f c \<in> carrier (relative_homology_group p Y T)) \<and> (\<forall>p<0. hom_induced p = (\<lambda>X S Y T f c. undefined))"
ap_spec_Nil_subset: fixes ap :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g" assumes "ap_spec ap" shows "set (ap gs bs ps [] data) \<subseteq> set ps"
i_append_eq_conv_conj: fixes xs :: "'b list"   and f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" shows "(xs \<frown> f = g) = (xs = g \<Down> length xs \<and> f = g \<Up> length xs)"
lens_indep_compat: fixes x :: "'b \<Longrightarrow> 'c"   and y :: "'d \<Longrightarrow> 'c" assumes "x \<bowtie> y" shows "x ##\<^sub>L y"
healthy_wp_Apply: fixes f :: "'a \<Rightarrow> 'a" shows "healthy (wp (Apply f))"
is_continuous: fixes U :: "'a set set" assumes "is_zariski_open U" shows "is_open (f \<^sup>\<inverse> X U)"
outs_\<I>_bot: shows "outs_\<I> \<bottom> = {}"
connected_space_eq: fixes X :: "'c topology" shows "connected_space X = (\<nexists>E1 E2. openin X E1 \<and> openin X E2 \<and> E1 \<union> E2 = topspace X \<and> E1 \<inter> E2 = {} \<and> E1 \<noteq> {} \<and> E2 \<noteq> {})"
ContractsWithUnilateralSubstitutesAndIRCAndLAD_axioms: shows "ContractsWithUnilateralSubstitutesAndIRCAndLAD Xd Xh (Pd_singletons_for_ds Y ds) Ch"
simps_1: fixes G :: "('a, 'b) graph"   and v :: "'a" assumes "valid_unSimpGraph G" shows "valid_unSimpGraph.adj_path G v [] = (v \<in> nodes G)"
\<psi>_mapsto: fixes b :: "'c"   and a :: "'c" assumes "C.ide b"   and "C.ide a" shows "\<psi> (b, a) \<in> Hom.set (b, a) \<rightarrow> C.hom b a"
pre_split_face_symI: fixes oldF :: "face"   and ram1 :: "nat"   and ram2 :: "nat"   and newVertexList :: "nat list" assumes "pre_split_face oldF ram1 ram2 newVertexList" shows "pre_split_face oldF ram2 ram1 newVertexList"
integrableI_nonneg: fixes f :: "'b \<Rightarrow> real"   and M :: "'b measure" assumes "f \<in> borel_measurable M"   and "AE x in M. 0 \<le> f x"   and "\<integral>\<^sup>+ x. ennreal (f x) \<partial>M < \<infinity>" shows "integrable M f"
TupleL: fixes t :: "trm"   and t' :: "trm"   and u :: "trm" assumes "t \<longmapsto> t'" shows "\<langle>t, u\<rangle> \<longmapsto> \<langle>t', u\<rangle>"
assms_2: fixes \<pi> :: "('state \<times> 'aprop set) stream" assumes "wfp AP' \<pi>"   and "\<pi>l \<noteq> [] \<Longrightarrow> stateOf \<pi> = stateOf (last \<pi>l)"   and "\<pi>l = [] \<Longrightarrow> stateOf \<pi> = s0"   and "F \<pi> \<pi>l" shows "\<chi>"
sums_suminf_ereal: fixes f :: "nat \<Rightarrow> real"   and x :: "real" assumes "f sums x" shows "(\<Sum>i. ereal (f i)) = ereal x"
sorted_sorted_list_of_set: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (linorder.sorted_list_of_set less_eq A)"
ab: shows "x = gcd x y * a \<and> y = gcd x y * b \<and> coprime a b"
elims_2: fixes x :: "('b ivl \<times> 'b) tree" assumes "inv_max_hi x" obtains "x = \<langle>\<rangle>"   | l :: "('b ivl \<times> 'b) tree" and a :: "'b ivl" and m :: "'b" and r :: "('b ivl \<times> 'b) tree" where "x = \<langle>l, (a, m), r\<rangle>" and "m = max3 a (max_hi l) (max_hi r) \<and> inv_max_hi l \<and> inv_max_hi r"
swap_involutive: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and zs :: "'varSort"   and x :: "'var"   and y :: "'var" assumes "good X" shows "X #[x \<and> y]_zs #[x \<and> y]_zs = X"