hcju/isabelleps huggingface dataset:
subset_Diff_insert: fixes A :: "'b set"   and B :: "'b set"   and x :: "'b"   and C :: "'b set" shows "(A \<subseteq> B - insert x C) = (A \<subseteq> B - C \<and> x \<notin> A)"
restrict_inv_into_Bij: fixes f :: "'c \<Rightarrow> 'c"   and S :: "'c set" assumes "f \<in> Bij S" shows "restrict (inv_into S f) S \<in> Bij S"
diff_le_self: fixes m :: "nat"   and n :: "nat" shows "m - n \<le> m"
cases: fixes x :: "('a, 'b) Term.term \<times> ('a, 'b) Term.term" obtains s :: "('a, 'b) Term.term" and t :: "('a, 'b) Term.term" where "x = (s, t)"
cross7_components_7: fixes x :: "(real, 7) vec"   and y :: "(real, 7) vec" shows "(x \<times>\<^sub>7 y) $ 7 = x $ 1 * y $ 3 - x $ 3 * y $ 1 + x $ 4 * y $ 5 - x $ 5 * y $ 4 + x $ 2 * y $ 6 - x $ 6 * y $ 2"
rev_bexI: fixes x :: "'f"   and A :: "'f set"   and P :: "'f \<Rightarrow> bool" assumes "x \<in> A"   and "P x" shows "\<exists>x\<in>A. P x"
S_ground_total: fixes F :: "('c \<times> nat) set"   and s :: "('c, 'd) Term.term"   and t :: "('c, 'd) Term.term" assumes "(pr_strict weights)\<^sup>=\<^sup>= = (pr_strict weights)\<^sup>=\<^sup>="   and "\<And>f g. \<lbrakk>f \<in> F; g \<in> F\<rbrakk> \<Longrightarrow> f = g \<or> pr_strict weights f g \<or> pr_strict weights g f"   and "funas_term s \<subseteq> F"   and "ground s"   and "funas_term t \<subseteq> F"   and "ground t" shows "s = t \<or> fst (kbo s t) \<or> fst (kbo t s)"
redT_updWs_None_SomeD: fixes t :: "'a"   and ws :: "'a \<Rightarrow> 'b wait_set_status option"   and was :: "('a, 'b) wait_set_action list"   and ws' :: "'a \<Rightarrow> 'b wait_set_status option"   and t' :: "'a"   and w :: "'b wait_set_status" assumes "redT_updWs t ws was ws'"   and "ws' t' = \<lfloor>w\<rfloor>"   and "ws t' = None" shows "t = t' \<and> (\<exists>w'. Suspend w' \<in> set was)"
z_def: shows "\<forall>x>z. Ifm (x # bs) (plusinf p) = Ifm (x # bs) p"
low_proc_0\<^sub>1_def: fixes p :: "prog" shows "low_proc_0\<^sub>1 p \<equiv> \<forall>\<sigma> u. match \<sigma> \<Gamma>\<^sub>0 \<and> the (\<sigma> ''lowIn'') = 0 \<longrightarrow> (\<exists>\<sigma>'. body p \<rhd> \<sigma> \<leadsto> Some \<sigma>' \<and> the (\<sigma>' ''lowOut'') = u)"
are_rules_2: fixes t :: "'b" shows "graph_rule (top_rule t)"
hom_lead_coeff: fixes p :: "'a mod_ring poly" shows "lead_coeff (to_int_poly p) = to_int_mod_ring (lead_coeff p)"
isomorphic_implies_ide_2: fixes f :: "'b"   and g :: "'b" assumes "f \<cong>\<^sub>D g" shows "D.ide g"
transform1: fixes P :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "'a state \<times> 'a state" assumes "P = true" shows "\<tau> \<Turnstile> P"
absorb_omega_exp2: fixes a :: "ordinal"   and b :: "ordinal" assumes "a < b" shows "\<omega> ** a + \<omega> ** b = \<omega> ** b"
UP_cring_def: fixes R :: "('a, 'b) ring_scheme" shows "UP_cring R \<equiv> cring R"
topology_inverse_iff: fixes U :: "'a set \<Rightarrow> bool" shows "istopology U = (openin (topology U) = U)"
rel_compp: fixes R1 :: "'f \<Rightarrow> 'k \<Rightarrow> bool"   and S1 :: "'k \<Rightarrow> 'i \<Rightarrow> bool"   and R2 :: "'g \<Rightarrow> 'l \<Rightarrow> bool"   and S2 :: "'l \<Rightarrow> 'j \<Rightarrow> bool" shows "Resumption.resumption.rel_resumption (R1 OO S1) (R2 OO S2) = Resumption.resumption.rel_resumption R1 R2 OO Resumption.resumption.rel_resumption S1 S2"
z_def: shows "z = normalize_quot (fst y', snd x')"
attach_compose: fixes conv1 :: "('i, 'j, 'm, 'n) converter"   and conv2 :: "('m, 'n, 'k, 'l) converter"   and res :: "('k, 'l) resource" shows "conv1 \<odot> conv2 \<rhd> res = conv1 \<rhd> conv2 \<rhd> res"
subst_typ_comp: fixes inst1 :: "((variable \<times> String.literal set) \<times> typ) list"   and inst2 :: "((variable \<times> String.literal set) \<times> typ) list"   and T :: "typ" shows "subst_typ inst1 (subst_typ inst2 T) = subst_typ (map (apsnd (subst_typ inst1)) inst2 @ inst1) T"
monad_optionT: fixes return :: "'a option \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a option \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad return bind" shows "monad (return_option return) (bind_option return bind)"
id: shows "(jordan_block n a ^\<^sub>m k) $$ (i, j) = (if i \<le> j then of_nat (k choose (j - i)) * a ^ (k + i - j) else (0::'a))"
simps_1: fixes e :: "('a, 'b, 'c) expr" shows "\<box> [e] = e"
vunion_vinsert_left: fixes a :: "V"   and A :: "V"   and B :: "V" shows "vinsert a A \<union>\<^sub>\<circ> B = vinsert a (A \<union>\<^sub>\<circ> B)"
fg: fixes x :: "'a" assumes "x \<in> topspace X" shows "f x = g x"
int_iffD2: fixes F :: "'a \<Rightarrow> bool"   and G :: "'a \<Rightarrow> bool" assumes "\<turnstile> F = G" shows "\<turnstile> G \<longrightarrow> F"
isnpolyh_mono: fixes n' :: "nat"   and n :: "nat"   and p :: "poly" assumes "n' \<le> n"   and "isnpolyh p n" shows "isnpolyh p n'"
oghoare_Await: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and F :: "'f set"   and P :: "('e, 'd, 'f) ann"   and b :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P (Await b c) Q, A" shows "\<exists>r P' Q' A'. P = AnnRec r P' \<and> \<Gamma>, \<Theta> \<tturnstile>\<^bsub>/F\<^esub> (r \<inter> b) P' c Q', A' \<and> atom_com c \<and> Q' \<subseteq> Q \<and> A' \<subseteq> A"
uadmit_sterm_ntadjoint: fixes \<sigma> :: "'b \<Rightarrow> ('sf, 'sz) trm"   and \<theta> :: "('sf + 'b, 'sz) trm"   and U :: "('sz + 'sz) set"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<omega> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and I :: "('sf, 'sc, 'sz) interp" assumes "NTUadmit \<sigma> \<theta> U"   and "Vagree \<nu> \<omega> (- U)"   and "\<And>i. dsafe (\<sigma> i)"   and "is_interp I" shows "sterm_sem (adjointFO I \<sigma> \<nu>) \<theta> = sterm_sem (adjointFO I \<sigma> \<omega>) \<theta>"
prod_list_le_mono: fixes us :: "'a list"   and vs :: "'a list" assumes "length us = length vs"   and "\<And>i. i < length vs \<Longrightarrow> (0::'a) \<le> us ! i \<and> us ! i \<le> vs ! i" shows "(0::'a) \<le> prod_list us \<and> prod_list us \<le> prod_list vs"
offending_flows_union_mono: fixes V :: "'a set"   and E :: "('a \<times> 'a) set"   and E' :: "('a \<times> 'a) set"   and nP :: "'a \<Rightarrow> 'b" assumes "wf_graph \<lparr>nodes = V, edges = E\<rparr>"   and "E' \<subseteq> E" shows "\<Union> (set_offending_flows \<lparr>nodes = V, edges = E'\<rparr> nP) \<subseteq> \<Union> (set_offending_flows \<lparr>nodes = V, edges = E\<rparr> nP)"
mult_pos_pos: fixes a :: "'b"   and b :: "'b" assumes "(0::'b) < a"   and "(0::'b) < b" shows "(0::'b) < a * b"
simps_1: fixes uu :: "'a extC" shows "(extC.Bot \<le> uu) = True"
aHom_add: fixes F :: "('e, 'f) aGroup_scheme"   and G :: "('g, 'h) aGroup_scheme"   and f :: "'e \<Rightarrow> 'g"   and a :: "'e"   and b :: "'e" assumes "aGroup F"   and "aGroup G"   and "f \<in> aHom F G"   and "a \<in> carrier F"   and "b \<in> carrier F" shows "f (a \<plusminus>\<^bsub>F\<^esub> b) = f a \<plusminus>\<^bsub>G\<^esub> f b"
check'_def: fixes dir :: "('i, 'a) Direction"   and x\<^sub>i :: "nat"   and s :: "('i, 'a) state" shows "check' dir x\<^sub>i s \<equiv> let l\<^sub>i = the (LB dir s x\<^sub>i); x\<^sub>j' = min_rvar_incdec dir s x\<^sub>i in case x\<^sub>j' of Inl I \<Rightarrow> set_unsat I s | Inr x\<^sub>j \<Rightarrow> pivot_and_update x\<^sub>i x\<^sub>j l\<^sub>i s"
idem: fixes a :: "'e multiset" shows "a \<union># a = a"
simps_2: fixes v :: "'a"   and u :: "bool"   and us :: "('a \<times> bool) list" shows "sat_list_to_bdt ((v, u) # us) = (if u then IF v (sat_list_to_bdt us) Falseif else IF v Falseif (sat_list_to_bdt us))"
rbt_contains_def: fixes a :: "'a"   and rbt :: "('a, 'b) RBT.rbt" shows "rbt_contains a rbt \<equiv> RBT.lookup rbt a \<noteq> None"
ts2t: shows "ts2 t = \<lfloor>xln'\<rfloor>"
hyp: fixes A :: "(('a, 'n) vec, 'm) vec" assumes "a \<noteq> i"   and "n < nrows A"   and "mod_type_class.to_nat a \<le> n" shows "row_add_iterate A n i j $ a $ b = row_add A a i (- A $ a $ j) $ a $ b"
elims_3: fixes x :: "'c \<Rightarrow> ('d \<times> 'e \<Rightarrow>f nat) option"   and xa :: "('c, 'f, 'g) new_thread_action list" assumes "\<not> thread_oks x xa" obtains ts :: "'c \<Rightarrow> ('d \<times> 'e \<Rightarrow>f nat) option" and ta :: "('c, 'f, 'g) new_thread_action" and tas :: "('c, 'f, 'g) new_thread_action list" where "x = ts" and "xa = ta # tas" and "\<not> (thread_ok ts ta \<and> thread_oks (redT_updT' ts ta) tas)"
Max_lem2: fixes pc :: "nat"   and bp :: "instr list"   and pc' :: "nat"   and s' :: "integer"   and s :: "integer" assumes "pc < length bp"   and "(pc', s') \<in> set (exec (bp ! pc) (pc, s))" shows "pc' \<le> Max (max_next_pcs bp)"
ss423conv: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "bijective x" shows "(x ; y \<le> z) = (y \<le> x\<^sup>T ; z)"
sets_restr_to_subalg: fixes M :: "'c measure"   and F :: "'c measure" assumes "subalgebra M F" shows "sets (restr_to_subalg M F) = sets F"
vector_2_2: fixes x :: "'a"   and y :: "'a" shows "vector [x, y] $ 2 = y"
old_set_rel_def: fixes R :: "('a \<times> 'b) set" shows "\<langle>R\<rangle>old_set_rel \<equiv> {(S, S'). S' = R `` S \<and> S \<subseteq> Domain R}"
Set_inter_code_1: fixes A'' :: "'b \<Rightarrow> bool"   and J :: "'b set" shows "Collect_set A'' \<inter> J = Collect_set (\<lambda>x. A'' x \<and> x \<in> J)"
bij_betw_arith_prog: fixes b :: "nat"   and a :: "int" assumes "0 < b" shows "bij_betw (\<lambda>n. a + int b * n) UNIV (arith_prog a b)"
abs_eq: fixes x :: "('a, 'b, 'c, 'd) FL_Formula.Tree" shows "FL_Formula.Not\<^sub>\<alpha> (FL_Formula.abs_Tree\<^sub>\<alpha> x) = FL_Formula.abs_Tree\<^sub>\<alpha> (FL_Formula.Tree.tNot x)"
USUP_true: fixes F :: "'b \<Rightarrow> 'a upred" shows "USUP F (\<lambda>P. true) = true"
x: shows "(\<forall>y. \<phi> y \<longrightarrow> P y x) \<and> (\<forall>y. P y x \<longrightarrow> (\<exists>z. \<phi> z \<and> O y z))"
cspan_as_span: fixes B :: "'d set" shows "cspan B = span (B \<union> (*\<^sub>C) \<i> ` B)"
sgn_pos: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "less zero a" shows "sgn a = one"
PPf_def2: fixes t1 :: "'b"   and t2 :: "'b" assumes "t1 \<in> trm"   and "t2 \<in> trm"   and "xx \<notin> FvarsT t1" shows "PPf t1 t2 = subst (subst Pf t1 yy) t2 xx"
inject_1: fixes x11 :: "'c \<times> 'd \<times> bool"   and x12 :: "'c option"   and y11 :: "'c \<times> 'd \<times> bool"   and y12 :: "'c option" shows "(Insert x11 x12 = Insert y11 y12) = (x11 = y11 \<and> x12 = y12)"
focus_Polys: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and X :: "'c set" assumes "p \<in> P[X]" shows "focus X p = (\<Sum>t\<in>keys p. monomial (monomial (lookup p t) 0) t)"
simps_13: fixes v :: "MIR.num" shows "qfree (Ge v) = True"
prems_1: shows "\<forall>i<length (t # T). \<exists>s. P ((t # T) ! i) s \<and> S' ! i = s \<cdot> \<delta>"
order_less_le_trans: fixes x :: "'c"   and y :: "'c"   and z :: "'c" assumes "x < y"   and "y \<le> z" shows "x < z"
N2: fixes A :: "nat set" assumes "A \<in> [N2]\<^bsup>4\<^esup>" shows "f2 A = j2"
WT_sub_gpvsD: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and gpv' :: "('c, 'a, 'b) gpv" assumes "\<I> \<turnstile>g gpv \<surd>"   and "gpv' \<in> sub_gpvs \<I> gpv" shows "\<I> \<turnstile>g gpv' \<surd>"
star_plus_loops: fixes x :: "'a" shows "x\<^sup>\<star> \<squnion> (1::'a) = x\<^sup>+ \<squnion> (1::'a)"
invar_set: fixes k :: "'a"   and v :: "real"   and l :: "'a kdt"   and r :: "'a kdt" shows "set_kdt (Node k v l r) = set_kdt l \<union> set_kdt r"
SN_on_trancl_imp_SN_on: fixes R :: "('a \<times> 'a) set"   and T :: "'a set" assumes "SN_on (R\<^sup>+) T" shows "SN_on R T"
Cons_4: shows "vars F \<union> vars (c # F'') \<subseteq> vars valuation"
arev_neq: fixes a :: "'b" assumes "a \<in> arcs I" shows "rev_I a \<noteq> a"
sel_2: fixes x21 :: "'a"   and x22 :: "('a, 'b) Term.term list" shows "the_Fun (Fun x21 x22) = x21"
dom_dom: fixes f :: "'a \<times> 'a \<times> 'a" assumes "VVV.arr f" shows "VVV.dom (VVV.dom f) = VVV.dom f"
C'01: shows "{0..1} \<subseteq> \<Union> C'"
string_to_nat_inv: fixes s :: "char list"   and n :: "nat" assumes "is_strong_nat_string s"   and "string_to_nat s = Some n" shows "nat_to_string n = s"
assms_9: shows "R_I = genericJoin I Q_I_pos Q_I_neg"
apartment_chamber_system_simplices: fixes X :: "'a set set" assumes "X \<in> \<A>" shows "order.greater_eq X (ChamberComplex.chamber_system X)"
l_inv_ex: fixes x :: "'a" assumes "x \<in> carrier C" shows "\<exists>y\<in>carrier C. y \<otimes>\<^bsub>C\<^esub> x = \<one>\<^bsub>C\<^esub>"
upred_arestr_lit: fixes v :: "bool"   and a :: "'c \<Longrightarrow> 'd" shows "\<guillemotleft>v\<guillemotright> \<restriction>\<^sub>p a = \<guillemotleft>v\<guillemotright>"
mgf_hyp: shows "\<forall>m. Suc m \<le> n \<longrightarrow> (\<forall>t. G,A\<turnstile>{=:m} t\<succ> {G\<rightarrow>})"
simps_1: fixes p :: "char list"   and ts :: "fterm list" shows "hlit_of_flit (Pos p ts) = Pos p (hterms_of_fterms ts)"
xSuct3: shows "x \<in> set (s_TS init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i + 1))"
sup_continuous_lfp: fixes F :: "'b \<Rightarrow> 'b" assumes "sup_continuous F" shows "\<nu> F = (\<Sqinter>i. (F ^^ i) \<top>)"
d: shows "type_max (to_total \<Gamma> x) mem\<^sub>2 = Low"
cf_pos_poly_eq_1: fixes f :: "int poly" shows "(cf_pos_poly f = 1) = (degree f = 0 \<and> f \<noteq> 0)"
less_bool_rel_def: shows "less_bool_rel \<equiv> {(x, y). x < y}"
children_eq_h2: fixes ptr' :: "(_) object_ptr"   and children :: "(_) node_ptr list" assumes "ptr' \<noteq> cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^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 shadow_root_ptr" shows "h2 \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children = h' \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children"
Let_eq_iff: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and \<Gamma>' :: "(Vars.var \<times> exp) list"   and e' :: "exp" shows "(Let \<Gamma> e = Let \<Gamma>' e') = ([map (\<lambda>x. atom (fst x)) \<Gamma>]lst. (e, \<Gamma>) = [map (\<lambda>x. atom (fst x)) \<Gamma>']lst. (e', \<Gamma>'))"
leg0_commutes: shows "HfHgh.cod.leg0 \<cdot> HfHgh.chine = HfHgh.dom.leg0"
asum_multTr_mp: fixes a :: "'a"   and m :: "int"   and n :: "int" assumes "a \<in> carrier A"   and "m < 0"   and "0 \<le> n" shows "aSum A (nat (- m)) (-\<^sub>a aSum A (nat n) a) = aSum A (nat (- m * n)) (-\<^sub>a a)"
linear_injective_imp_surjective: fixes f :: "'b \<Rightarrow> 'c" assumes "linear (*a) (*b) f"   and "inj f"   and "vs2.dim UNIV = vs1.dim UNIV" shows "surj f"
add_cancel_left_right: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and a :: "'g"   and b :: "'g" assumes "class.cancel_comm_monoid_add plus minus zero" shows "(plus a b = a) = (b = zero)"
DFS_axioms: shows "DFS G (Tarjan_def.tarjan_params G)"
the_\<theta>_props_2: shows "local.iso the_\<theta>"
homhc: shows "homotopic_with_canon (\<lambda>z. True) (sphere (0::'a) 1 \<inter> S) (sphere (0::'a) 1 \<inter> T) h (\<lambda>x. c)"
runit_ide_eq: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b arrow_of_spans_data" assumes "span_bicategory C prj0 prj1"   and "partial_magma.ide (span_vertical_category.vcomp C) f" shows "bicategory.runit (span_vertical_category.vcomp C) (span_bicategory.hcomp C prj0 prj1) (span_bicategory.assoc C prj0 prj1) (span_bicategory.unit C prj0 prj1) (span_vertical_category.src C) f = \<lparr>Chn = prj1 (Leg0 (Dom f)) (partial_magma.cod C (Leg0 (Dom f))), Dom = \<lparr>Leg0 = prj0 (Leg0 (Dom f)) (partial_magma.cod C (Leg0 (Dom f))), Leg1 = C (Leg1 (Dom f)) (prj1 (Leg0 (Dom f)) (partial_magma.cod C (Leg0 (Dom f))))\<rparr>, Cod = Cod f\<rparr>"
i_def: shows "i = Suc (Suc (Suc (Suc (Suc (Suc 0)))))"
hom: fixes g :: "'g"   and g' :: "'g" assumes "g \<in> UNIV"   and "g' \<in> UNIV" shows "T (g + g') = T g + T g'"
of_bool_def: fixes one :: "'a"   and zero :: "'a"   and p :: "bool" assumes "class.zero_neq_one one zero" shows "zero_neq_one.of_bool one zero p = (if p then one else zero)"
wf_until_aux_Cons3: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and n :: "nat"   and R :: "'a list set"   and pos :: "bool"   and \<phi> :: "'a MFOTL.formula"   and I :: "\<I>"   and \<psi> :: "'a MFOTL.formula"   and t :: "nat"   and a1 :: "'a option list set"   and a2 :: "'a option list set"   and aux :: "(nat \<times> 'a option list set \<times> 'a option list set) list"   and ne :: "nat" assumes "wf_until_aux \<sigma> n R pos \<phi> I \<psi> ((t, a1, a2) # aux) ne" shows "qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. \<exists>j\<ge>ne. j < Suc (ne + length aux) \<and> mem (\<tau> \<sigma> j - \<tau> \<sigma> ne) I \<and> MFOTL.sat \<sigma> (map the v) j \<psi> \<and> (\<forall>k\<in>{ne..<j}. if pos then MFOTL.sat \<sigma> (map the v) k \<phi> else \<not> MFOTL.sat \<sigma> (map the v) k \<phi>)) a2"
Nil: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and s :: "('e, 'f) xstate" shows "\<Gamma>\<turnstile>[],[] \<Down> s"
second_vertices_in_V: shows "Q.second_vertices \<subseteq> V\<^bsub>H_x\<^esub>"
of_nat_fract: fixes k :: "nat" shows "of_nat k = Fract (of_nat k) (1::'b)"
pseudofunctor_axioms: shows "pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C G \<Phi>\<^sub>G"
supporting_hyperplane_closed_point: fixes S :: "'b set"   and z :: "'b" assumes "convex S"   and "closed S"   and "S \<noteq> {}"   and "z \<notin> S" shows "\<exists>a b. \<exists>y\<in>S. a \<bullet> z < b \<and> a \<bullet> y = b \<and> (\<forall>x\<in>S. b \<le> a \<bullet> x)"
expand_prop_exist: fixes n_ns :: "'b node \<times> 'b node set"   and \<xi> :: "nat \<Rightarrow> 'b set" shows "expand n_ns \<le> SPEC (\<lambda>r. expand_assm_exist \<xi> n_ns \<longrightarrow> expand_rslt_exist \<xi> n_ns r)"
continuous_on_sgn: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_on s f"   and "\<forall>x\<in>s. f x \<noteq> (0::'d)" shows "continuous_on s (\<lambda>x. sgn (f x))"
Pg_gen_vp: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "Vr K v \<diamondsuit>\<^sub>p Pg K v = vp K v"
lookup_plus_fun: fixes s :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b" shows "lookup (s + t) = lookup s + lookup t"
rComm1: fixes \<Psi>\<^sub>Q :: "'b"   and M :: "'a"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and K :: "'a"   and xvec :: "name list"   and Q' :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q \<turnstile> M \<leftrightarrow> K"   and "distinct xvec"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* N"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* xvec"   and "A\<^sub>P \<sharp>* Q'"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* K"   and "A\<^sub>Q \<sharp>* N"   and "A\<^sub>Q \<sharp>* P'"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* xvec"   and "A\<^sub>Q \<sharp>* Q'"   and "A\<^sub>Q \<sharp>* C"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* \<Psi>\<^sub>P"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* K"   and "xvec \<sharp>* Q"   and "xvec \<sharp>* \<Psi>\<^sub>Q"   and "xvec \<sharp>* C"   and "x\<alpha> = \<tau>"   and "xT = \<lparr>\<nu>*xvec\<rparr>P' \<parallel> Q'" shows "Prop"
flow_undefined0: fixes t :: "real"   and t0 :: "real"   and x :: "'a" assumes "t - t0 \<notin> existence_ivl0 x" shows "flow0 x (t - t0) = (0::'a)"
indefinite_integral_continuous_1: fixes f :: "real \<Rightarrow> 'b"   and a :: "real"   and b :: "real" assumes "f integrable_on {a..b}" shows "continuous_on {a..b} (\<lambda>x. integral {a..x} f)"
to_nat_from_nat_id: fixes i :: "nat" assumes "i < CARD('a)" shows "Bij_Nat.to_nat (Bij_Nat.from_nat i) = i"
i_Intersect: fixes S :: "'a set set" assumes "\<And>I. I \<in> S \<Longrightarrow> ideal I R"   and "S \<noteq> {}" shows "ideal (\<Inter> S) R"
s: shows "(\<Sum>x\<in>UNIV. s x *s X $ x) = v"
Join_stable: fixes F :: "'b program"   and G :: "'b program"   and A :: "'b set" shows "(F \<squnion> G \<in> stable A) = (F \<in> stable A \<and> G \<in> stable A)"
while_top_def: fixes b :: "'b upred"   and P :: "'b hrel" shows "while\<^sup>\<top> b do P od = (\<nu> X \<bullet> P ;; X \<triangleleft> b \<triangleright>\<^sub>r II)"
exhaust_sel: fixes sum :: "'b + 'c" obtains "sum = Inl (projl sum)"   | "sum = Inr (projr sum)"
low_Eval: fixes e :: "Expr"   and s :: "Var \<Rightarrow> Val"   and t :: "Var \<Rightarrow> Val" assumes "\<And>x. x \<in> EVars e \<Longrightarrow> CONTEXT x = low"   and " s \<approx> t " shows "evalE e s = evalE e t"
\<theta>: shows "\<theta> \<equiv> \<lambda>y. if x = y then \<delta> y else \<sigma> y"
convex_set_plus: fixes S :: "'c set"   and T :: "'c set" assumes "convex S"   and "convex T" shows "convex (S + T)"
topological_basisI: fixes "open" :: "'c set \<Rightarrow> bool"   and B :: "'c set set" assumes "class.topological_space open"   and "\<And>B'. B' \<in> B \<Longrightarrow> open B'"   and "\<And>O' x. \<lbrakk>open O'; x \<in> O'\<rbrakk> \<Longrightarrow> \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'" shows "topological_space.topological_basis open B"
squarefreeI: fixes n :: "'a" assumes "\<And>x. x\<^sup>2 dvd n \<Longrightarrow> x dvd (1::'a)" shows "squarefree n"
removeAll_filter_not: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" assumes "\<not> P x" shows "removeAll x (filter P xs) = filter P xs"
intros: fixes P :: "'a prog"   and C :: "String.literal"   and D :: "String.literal"   and rest :: "fdecl list \<times> 'a option mdecl list" assumes "class P C = \<lfloor>(D, rest)\<rfloor>"   and "C \<noteq> Object" shows "P \<turnstile> C \<prec>\<^sup>1 D"
hyps: fixes i :: "nat" shows "i < row_length M * row_length M2 \<and> mat (row_length M) (length M) M \<and> mat (row_length M2) (length M2) M2 \<longrightarrow> row (M \<otimes> M2) i = vec_vec_Tensor (row M (i div row_length M2)) (row M2 (i mod row_length M2))"
vfsequence_def: fixes xs :: "V" shows "vfsequence xs \<equiv> vsv xs \<and> vfsequence_axioms xs"
F_inf_nS: fixes s :: "'s"   and f :: "'s \<Rightarrow> ennreal" assumes "s \<notin> S" shows "F_inf f s = undefined"
inv_approx_def: fixes E :: "'b set set"   and S :: "'b set"   and X :: "'b set"   and r :: "nat" assumes "Graph_E E" shows "Graph_E.inv_approx E S X r = (Graph_E.inv_iv E S X \<and> card X \<le> card S * r)"
trans: fixes b :: "'d multiset"   and a :: "'d multiset"   and c :: "'d multiset" assumes "b \<subseteq># a"   and "c \<subseteq># b" shows "c \<subseteq># a"
hyps_6: shows "bn \<alpha> \<sharp>* subject \<alpha>"
ang_vec_a_def: fixes z1 :: "complex"   and z2 :: "complex" shows "\<angle>a z1 z2 \<equiv> acute_ang (\<angle>c z1 z2)"