hcju/isabelleps huggingface dataset:
abs_eq: fixes x :: "'b \<times> 'b" assumes "fractrel x x" shows "- abs_fract x = abs_fract (- fst x, snd x)"
insert_idem_2: shows "insert t0 (existence_ivl t0 x0 \<inter> {t0..}) = existence_ivl t0 x0 \<inter> {t0..}"
raw_match_term_dom_res_subset_tvs: fixes t :: "Core.term"   and u :: "Core.term"   and subs :: "variable \<times> String.literal set \<Rightarrow> typ option"   and subs' :: "variable \<times> String.literal set \<Rightarrow> typ option" assumes "raw_match_term t u subs = Some subs'" shows "dom subs' \<subseteq> tvs t \<union> dom subs"
S4a_live: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> \<box>(ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<and> (\<forall>l. $MemInv mm l)) \<and> WF(RNext rmCh mm ires p)_m p \<longrightarrow> (S4 rmhist p \<and> ires!p = #NotAResult \<leadsto> S4 rmhist p \<and> ires!p \<noteq> #NotAResult \<or> S5 rmhist p)"
D: shows "\<langle>n, Key (Auth_ShaKey n)\<rangle> \<in> spied s"
assms_2: fixes X :: "nat set" assumes "X \<subseteq> {..n}" shows "sum f X \<le> sum g (R `` X)"
gcd_dvd_some_gcd: fixes a :: "'a"   and b :: "'a" shows "gcd a b dvd some_gcd a b"
currentD_finite_OUT: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v" assumes "current \<Gamma> f" shows "d_OUT f x \<noteq> \<top>"
projs_5: fixes dsn :: "nat"   and dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat" shows "\<pi>\<^sub>6 (dsn, dsk, flag, hops, nhip) = nhip"
terminates_Normal_elim_cases_7: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and b :: "'e set"   and c :: "('e, 'd, 'f) com"   and s :: "'e" assumes "\<Gamma>\<turnstile>While b c \<down> Normal s" obtains "s \<in> b" and "\<Gamma>\<turnstile>c \<down> Normal s" and "\<forall>s'. \<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow> s' \<longrightarrow> \<Gamma>\<turnstile>While b c \<down> s'"   | "s \<notin> b"
mincsk: shows "min (length cs) (Suc k) = Suc k"
dist_triangle3: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and a :: "'a" assumes "class.metric_space dist uniformity open" shows "dist x y \<le> dist a x + dist a y"
vertex_subtree_connected: fixes v :: "'a"   and s :: "'c"   and t :: "'c"   and xs :: "'c list" assumes "v \<in> V"   and "s \<in> vertex_subtree v"   and "t \<in> vertex_subtree v"   and "s \<leadsto>xs\<leadsto>\<^bsub>T\<^esub> t" shows "set xs \<subseteq> vertex_subtree v"
perm_1: fixes pi :: "(name \<times> name) list"   and x2 :: "'d"   and x1 :: "('d, 'e, 'f) psi" shows "pi \<bullet> x2 \<prec>' x1 = (pi \<bullet> x2) \<prec>' (pi \<bullet> x1)"
mult_pos_neg: fixes a :: "'a"   and b :: "'a" assumes "(0::'a) < a"   and "b < (0::'a)" shows "a * b < (0::'a)"
arr_Set_ArrCod_in_Vset: shows "T\<lparr>ArrCod\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
\<phi>_eq: shows "\<phi> = (\<lambda>\<mu>. if \<guillemotleft>\<mu> : src e\<^sub>0 \<rightarrow> src e\<^sub>1\<guillemotright> then ((d\<^sub>1 \<star> \<a>\<^sup>-\<^sup>1[e\<^sub>1, cod \<mu> \<star> d\<^sub>0, e\<^sub>0]) \<cdot> \<a>[d\<^sub>1, e\<^sub>1, (cod \<mu> \<star> d\<^sub>0) \<star> e\<^sub>0] \<cdot> ((d\<^sub>1 \<star> e\<^sub>1) \<star> \<a>\<^sup>-\<^sup>1[cod \<mu>, d\<^sub>0, e\<^sub>0]) \<cdot> (\<eta>\<^sub>1 \<star> cod \<mu> \<star> \<eta>\<^sub>0) \<cdot> \<l>\<^sup>-\<^sup>1[cod \<mu> \<star> src e\<^sub>0] \<cdot> \<r>\<^sup>-\<^sup>1[cod \<mu>]) \<cdot> \<mu> else null)"
shconf_hnew: fixes P :: "'a prog"   and h :: "heap"   and sh :: "sheap"   and a :: "nat"   and obj :: "obj" assumes "P,h \<turnstile>\<^sub>s sh \<surd>"   and "h a = None" shows "P,h(a \<mapsto> obj) \<turnstile>\<^sub>s sh \<surd>"
assms_5: fixes X :: "'a" assumes "X \<in> obj\<^bsub>NTCatDom \<eta>1\<^esub>" shows "\<eta>1 $$ X = \<eta>2 $$ X"
add_left_cancel: fixes a :: "'ag"   and b :: "'ag"   and c :: "'ag" assumes "a \<in> U"   and "b \<in> U"   and "c \<in> U" shows "(a +\<^sub>o\<^sub>w b = a +\<^sub>o\<^sub>w c) = (b = c)"
proj_append_2: fixes l :: "'e"   and A :: "('e strand_label \<times> 'd) list"   and B :: "('e strand_label \<times> 'd) list" shows "proj_unl l (A @ B) = proj_unl l A @ proj_unl l B"
continuous_on_transfer_right_total: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and AP :: "'b \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "Domainp A = AP"   and "bi_unique A"   and "right_total A"   and "rel_fun (rel_set A) (=) (openin (top_of_set (Collect AP))) open"   and "bi_unique B"   and "bi_total B"   and "rel_fun (rel_set B) (=) open open" shows "rel_fun (rel_set A) (rel_fun (rel_fun A B) (=)) (\<lambda>X. continuous_on (X \<inter> Collect AP)) continuous_on"
sep_add_zero_sym: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.sep_algebra plus zero sep_disj" shows "plus zero x = x"
eq_nextl_refines_eq_app_right: shows "eq_nextl \<subseteq> eq_app_right language"
assms_9: fixes a :: "name"   and x :: "name"   and P' :: "pi"   and P'' :: "pi"   and C :: "'a" assumes "P \<longmapsto> a<\<nu>x> \<prec> P'"   and "!P \<longmapsto> a<x> \<prec> P''"   and "x \<sharp> P"   and "x \<sharp> C"   and "\<And>C. F C (!P) (a<x> \<prec> P'')" shows "F C (P \<parallel> !P) (\<tau> \<prec> <\<nu>x>(P' \<parallel> P''))"
basis_subset_carrier: fixes X :: "'a set" assumes "\<E>.basis X" shows "X \<subseteq> \<E>"
type_definition_downset: shows "type_definition Rep_downset Abs_downset (range \<down>)"
gcd_proj1_if_dvd_nat: fixes x :: "nat"   and y :: "nat" assumes "x dvd y" shows "gcd x y = x"
gtrp_env: fixes s :: "'a"   and sl :: "'a"   and c :: "'a multiset"   and e :: "'b"   and s' :: "'a"   and c' :: "'a multiset"   and S :: "(('a \<times> 'a multiset) \<times> 'b \<times> 'a \<times> 'a multiset) set" assumes "((s, add_mset sl c), e, s', add_mset sl c') \<in> S" shows "((sl, add_mset s c), ENV e, sl, add_mset s' c') \<in> gtrp S"
limi_mono: fixes A :: "'a set"   and B :: "'a set"   and g :: "'b" assumes "A \<subseteq> B" shows "limi A g \<subseteq> limi B g"
Var: fixes env :: "Lambda_Example.type list"   and x :: "nat"   and T :: "Lambda_Example.type" assumes "nth_el1 env x T" shows "env \<turnstile> Var x : T"
FP_equivalence: fixes F :: "'b program" shows "FP F = FP_Orig F"
gt_embedding_property: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   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 t :: "('b, 'a) tm"   and s :: "('b, 'a) tm" assumes "epo ground_heads_var gt_sym arity_sym arity_var extf"   and "t \<unrhd>\<^sub>e\<^sub>m\<^sub>b s"   and "t \<noteq> s" shows "epo.gt ground_heads_var gt_sym extf t s"
assms_1: fixes a :: "'a" assumes "a \<in> A" shows "{a} \<in> M"
induct: fixes P :: "atom fm \<Rightarrow> bool"   and a0 :: "atom fm" assumes "P TrueF"   and "P FalseF"   and "\<And>A B. \<lbrakk>P A; P B\<rbrakk> \<Longrightarrow> P (And A B)"   and "\<And>A B. \<lbrakk>P A; P B\<rbrakk> \<Longrightarrow> P (Or A B)"   and "\<And>A. P A \<Longrightarrow> P (Neg A)"   and "\<And>A. P (fm.Atom A)"   and "\<And>A. P (ExN 2 A) \<Longrightarrow> P (ExQ (ExQ A))"   and "\<And>j A. P (ExN (j + 1) A) \<Longrightarrow> P (ExQ (ExN j A))"   and "\<And>j A. P (ExN (j + 1) A) \<Longrightarrow> P (ExN j (ExQ A))"   and "\<And>i j A. P (ExN (i + j) A) \<Longrightarrow> P (ExN i (ExN j A))"   and "P TrueF \<Longrightarrow> P (ExQ TrueF)"   and "P FalseF \<Longrightarrow> P (ExQ FalseF)"   and "\<And>v. P (fm.Atom v) \<Longrightarrow> P (ExQ (fm.Atom v))"   and "\<And>v va. P (And v va) \<Longrightarrow> P (ExQ (And v va))"   and "\<And>v va. P (Or v va) \<Longrightarrow> P (ExQ (Or v va))"   and "\<And>v. P (Neg v) \<Longrightarrow> P (ExQ (Neg v))"   and "\<And>v. P (AllQ v) \<Longrightarrow> P (ExQ (AllQ v))"   and "\<And>v va. P (AllN v va) \<Longrightarrow> P (ExQ (AllN v va))"   and "\<And>A. P (AllN 2 A) \<Longrightarrow> P (AllQ (AllQ A))"   and "\<And>j A. P (AllN (j + 1) A) \<Longrightarrow> P (AllQ (AllN j A))"   and "\<And>j A. P (AllN (j + 1) A) \<Longrightarrow> P (AllN j (AllQ A))"   and "\<And>i j A. P (AllN (i + j) A) \<Longrightarrow> P (AllN i (AllN j A))"   and "P TrueF \<Longrightarrow> P (AllQ TrueF)"   and "P FalseF \<Longrightarrow> P (AllQ FalseF)"   and "\<And>v. P (fm.Atom v) \<Longrightarrow> P (AllQ (fm.Atom v))"   and "\<And>v va. P (And v va) \<Longrightarrow> P (AllQ (And v va))"   and "\<And>v va. P (Or v va) \<Longrightarrow> P (AllQ (Or v va))"   and "\<And>v. P (Neg v) \<Longrightarrow> P (AllQ (Neg v))"   and "\<And>v. P (ExQ v) \<Longrightarrow> P (AllQ (ExQ v))"   and "\<And>v va. P (ExN v va) \<Longrightarrow> P (AllQ (ExN v va))"   and "\<And>j. P (AllN j TrueF)"   and "\<And>j. P (AllN j FalseF)"   and "\<And>j v. P (AllN j (fm.Atom v))"   and "\<And>j v va. P (AllN j (And v va))"   and "\<And>j v va. P (AllN j (Or v va))"   and "\<And>j v. P (AllN j (Neg v))"   and "\<And>j v. P (AllN j (ExQ v))"   and "\<And>j v va. P (AllN j (ExN v va))"   and "\<And>j. P (ExN j TrueF)"   and "\<And>j. P (ExN j FalseF)"   and "\<And>j v. P (ExN j (fm.Atom v))"   and "\<And>j v va. P (ExN j (And v va))"   and "\<And>j v va. P (ExN j (Or v va))"   and "\<And>j v. P (ExN j (Neg v))"   and "\<And>j v. P (ExN j (AllQ v))"   and "\<And>j v va. P (ExN j (AllN v va))" shows "P a0"
rule_dom: shows "(N, \<alpha> @ \<beta>) \<in> \<RR>"
Contra': fixes A :: "fmla"   and H :: "fmla set" assumes "insert A H \<turnstile> neg A" shows "H \<turnstile> neg A"
mapl_G_def: fixes l1 :: "'l \<Rightarrow> 's"   and l2 :: "'m \<Rightarrow> 't" shows "mapl_G l1 l2 = map_G l1 l2 id id id id"
blinfun_eqI: fixes x :: "'a \<Rightarrow>\<^sub>L 'b"   and y :: "'a \<Rightarrow>\<^sub>L 'b" assumes "\<And>i. x *\<^sub>v i = y *\<^sub>v i" shows "x = y"
min: fixes y :: "'a" assumes "r y z" shows "\<not> (P y \<and> r\<^sup>*\<^sup>* y x)"
csum_absorb2': fixes r2 :: "('c \<times> 'c) set"   and r1 :: "('d \<times> 'd) set" assumes "Card_order r2"   and "r1 \<le>o r2"   and "cinfinite r1 \<or> cinfinite r2" shows "r1 +c r2 =o r2"
elims: fixes x :: "('b \<times> 'b) list"   and xa :: "'b"   and y :: "'b" assumes "preprm_apply x xa = y" obtains xb :: "'b" where "x = []" and "xa = xb__" and "y = xb__"   | s :: "'b \<times> 'b" and ss :: "('b \<times> 'b) list" and xb :: "'b" where "x = s # ss" and "xa = xb" and "y = swp_apply s (preprm_apply ss xb)"
weakly_equivalent_iff_not_distinguished: fixes \<tau> :: "'b"   and satisfies :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and transition :: "'c \<Rightarrow> ('b, 'c) residual \<Rightarrow> bool"   and P :: "'c"   and Q :: "'c" assumes "indexed_weak_nominal_ts TYPE('a) \<tau> satisfies transition" shows "indexed_weak_nominal_ts.weakly_logically_equivalent TYPE('a) \<tau> satisfies transition P Q = (\<nexists>x. indexed_weak_nominal_ts.weak_formula \<tau> x \<and> indexed_weak_nominal_ts.is_distinguishing_formula satisfies transition x P Q)"
map_of_Cons_code_1: fixes k :: "'m" shows "map_of [] k = None"
uminus_SUP: fixes B :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "- Sup (B ` A) = (INF x\<in>A. - B x)"
rep2: shows "[t2]<>v \<equiv>\<^sub>P r2"
index_oddpart_2: fixes m :: "nat"   and i :: "nat" assumes "odd m" shows "oddpart (2 ^ i * m) = m"
bisim_simple_\<R>\<^sub>u: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and bexp_neg :: "'d \<Rightarrow> 'd"   and assign_post :: "'d set \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'd set"   and dma_type :: "'a \<Rightarrow> 'd set"   and pred_False :: "'d"   and lc :: "(('a, 'c, 'd) Stmt \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)"   and lc' :: "(('a, 'c, 'd) Stmt \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)"   and \<Gamma> :: "'a \<Rightarrow> 'd set option"   and \<S> :: "'a set \<times> 'a set"   and P :: "'d set" assumes "sifum_types ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type pred_False"   and "(lc, lc') \<in> sifum_types.\<R> ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type \<Gamma> \<S> P" shows "fst (fst lc) = fst (fst lc')"
prod_rel_simp: fixes a :: "'a"   and b :: "'b"   and a' :: "'c"   and b' :: "'d"   and R1 :: "('a \<times> 'c) set"   and R2 :: "('b \<times> 'd) set" shows "(((a, b), a', b') \<in> R1 \<times>\<^sub>r R2) = ((a, a') \<in> R1 \<and> (b, b') \<in> R2)"
Deduct2_axioms: shows "Deduct2 var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv bprv"
pseudominimal_inD1: fixes P :: "'b set"   and x :: "'b" assumes "order.pseudominimal_in P x" shows "x \<in> P"
int_clset_def: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and S :: "'b eclause set" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.int_clset trm_ord sel S = basic_superposition.same_values (\<lambda>x. basic_superposition.trm_rep trm_ord sel x S)"
prems_1: shows "insert z0 X \<subseteq> interior S"
Nmlize_Hcomp_Hcomp': fixes t :: "'a local.term"   and u :: "'a local.term"   and v :: "'a local.term" assumes "Arr t"   and "Arr u"   and "Arr v"   and "Src t = Trg u"   and "Src u = Trg v" shows "\<^bold>\<lfloor>t \<^bold>\<star> u \<^bold>\<star> v\<^bold>\<rfloor> = \<^bold>\<lfloor>\<^bold>\<lfloor>t\<^bold>\<rfloor> \<^bold>\<star> \<^bold>\<lfloor>u\<^bold>\<rfloor> \<^bold>\<star> \<^bold>\<lfloor>v\<^bold>\<rfloor>\<^bold>\<rfloor>"
terminating_iff1: fixes x :: "'a" assumes "path x" shows "terminating x = (has_start_end_points x \<or> x = 0)"
apartment_chamberD_simplex: fixes X :: "'a set set"   and x :: "'a set" assumes "X \<in> \<A>"   and "SimplicialComplex.maxsimp X x" shows "x \<in> X"
MM6: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> \<box>F \<or> \<box>G \<longrightarrow> \<box>(\<box>F \<or> \<box>G)"
lunit_char_1: fixes f :: "'a" assumes "ide f" shows "\<guillemotleft>\<l>[f] : L f \<Rightarrow> f\<guillemotright>"
shiftl_eq_push_bit: fixes a :: "'a"   and n :: "nat" shows "a << n = push_bit n a"
psi_is_psi_l0: shows "\<psi> = psi_l 0"
none_case: fixes r :: "'a" assumes "s r = None" shows "s' (\<alpha> r) = None"
cgood_def: fixes ct :: "('sequent, 'rule, 'link) ctree" shows "cgood ct \<equiv> good (treeOf pointsTo ct)"
filter_pdevs_cong: fixes x :: "'a pdevs"   and y :: "'a pdevs"   and P :: "nat \<Rightarrow> 'a \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> 'a \<Rightarrow> bool" assumes "x = y"   and "\<And>i. i \<in> pdevs_domain y \<Longrightarrow> P i (pdevs_apply x i) = Q i (pdevs_apply y i)" shows "filter_pdevs P x = filter_pdevs Q y"
sup_inf_absorb: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.lattice inf less_eq less sup" shows "sup x (inf x y) = x"
column_add_def: fixes A :: "(('a, 'b) vec, 'c) vec"   and n :: "'b"   and m :: "'b"   and q :: "'a" shows "column_add A n m q = (\<chi>i j. if j = n then A $ i $ n + A $ i $ m * q else A $ i $ j)"
left_gpv_bind_gpv: fixes gpv :: "('f, 'b, 'd) gpv"   and f :: "'f \<Rightarrow> ('a, 'b, 'd) gpv" shows "left_gpv (gpv \<bind> f) = left_gpv gpv \<bind> left_gpv \<circ> f"
wf_chains: fixes G :: "diagram"   and \<Pi> :: "(node \<rightharpoonup>\<^sub>f topbot, node + node fset \<times> command_gadget \<times> node fset) chain" assumes "wf_dia G"   and "\<Pi> \<in> ps_chains G" shows "wf_ps_chain \<Pi> \<and> post \<Pi> = [ terminals G |=> Bot ]"
no_cycles: fixes G :: "('c, 'd) Graph_scheme"   and xs :: "'c list" assumes "Tree G" shows "\<not> Graph.cycle G xs"
Qring_fix1: fixes R :: "('d, 'e) Ring_scheme"   and a :: "'d"   and I :: "'d set" assumes "Ring R"   and "a \<in> carrier R"   and "ideal R I"   and "a \<in> I" shows "a \<uplus>\<^bsub>R\<^esub> I = I"
simps_4: fixes \<phi> :: "('a, 'b, 'c) formula"   and \<psi> :: "('a, 'b, 'c) formula"   and \<sigma> :: "('a, 'b, 'c) subst" shows "Fsubst (\<phi> && \<psi>) \<sigma> = (Fsubst \<phi> \<sigma> && Fsubst \<psi> \<sigma>)"
oneTS_stepx: fixes x :: "'a"   and y :: "'a"   and hs :: "'a list" assumes "x \<noteq> y" shows "TS_step_d ([x, y], x # x # hs) y = ((0, []), y # x # x # hs)"
unsigned_of_int: fixes k :: "int" shows "unsigned (word_of_int k) = of_nat (nat (take_bit LENGTH('c) k))"
reachable_rtranclI: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and v :: "'c" assumes "u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v" shows "(u, v) \<in> (arcs_ends G)\<^sup>*"
distinct_11: fixes x3 :: "(Term_Class.term \<times> sterm) list"   and x41 :: "sterm"   and x42 :: "sterm" shows "Sabs x3 \<noteq> x41 $\<^sub>s x42"
HEndPhase1_HInv1: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HInv1 s"   and "HEndPhase1 s s' p" shows "HInv1 s'"
reflexive: fixes a :: "'a" assumes "a \<in> R" shows "(a, a) \<in> Ring_Congruence"
pderivs_lang_def: fixes A :: "'b list set"   and r :: "'b rexp" shows "pderivs_lang A r \<equiv> \<Union>x\<in>A. pderivs x r"
word_less_two_pow_divI: fixes x :: "'a word"   and n :: "nat"   and m :: "nat" assumes "x < 2 ^ (n - m)"   and "m \<le> n"   and "n < LENGTH('a)" shows "x < 2 ^ n div 2 ^ m"
aux2: fixes a' :: "bool list" shows "nlists UNIV \<eta> \<inter> {x. x \<noteq> a'} \<noteq> {}"
minus_eq: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a" shows "x \<ominus>\<^bsub>P\<^esub> y = x \<oplus>\<^bsub>P\<^esub> \<ominus>\<^bsub>P\<^esub> y"
le_trans_trans: fixes x :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and y :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and z :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "le_trans x y"   and "le_trans y z" shows "le_trans x z"
assms_2: shows "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"
trms_of_deg_leq_deg: fixes f :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "f \<in> carrier P" shows "deg R (trms_of_deg_leq k f) \<le> k"
distinct_1: fixes x11 :: "'a"   and x12 :: "bool"   and x2 :: "bool" shows "<x11, inM x12> \<noteq> <\<bottom>, outM x2>"
drop_update_cancel: fixes n :: "nat"   and m :: "nat"   and xs :: "'d list"   and x :: "'d" assumes "n < m" shows "drop m (xs[n := x]) = drop m xs"
signed_take_bit_decr_length_iff: fixes k :: "'k"   and l :: "'k" shows "(signed_take_bit (LENGTH('l) - Suc 0) k = signed_take_bit (LENGTH('l) - Suc 0) l) = (take_bit LENGTH('l) k = take_bit LENGTH('l) l)"
elims_3: fixes x :: "atom set \<times> 'a"   and xa :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xb :: "'a \<Rightarrow> atom set"   and xc :: "perm"   and xd :: "atom set \<times> 'a" assumes "\<not> x \<approx>set xa xb xc xd" obtains bs :: "atom set" and xe :: "'a" and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and f :: "'a \<Rightarrow> atom set" and p :: "perm" and cs :: "atom set" and y :: "'a" where "x = (bs, xe)" and "xa = R" and "xb = f" and "xc = p" and "xd = (cs, y)" and "\<not> (f xe - bs = f y - cs \<and> (f xe - bs) \<sharp>* p \<and> R (p \<bullet> xe) y \<and> p \<bullet> bs = cs)"
int_pow_distrib: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and y :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and i :: "int" assumes "x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))"   and "y \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))" shows "(x \<otimes>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> y) [^]\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> i = x [^]\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> i \<otimes>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> y [^]\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> i"
subint1D: fixes I :: "qtname"   and J :: "qtname"   and G :: "prog" assumes "G\<turnstile>I\<prec>I1J" shows "? i:iface G I: J \<in> set (isuperIfs i)"
cod_S': fixes A :: "type_scheme list"   and t :: "typ"   and S :: "nat \<Rightarrow> typ" shows "cod (\<lambda>n. if n \<in> free_tv A \<union> free_tv t then S n else TVar n) \<subseteq> free_tv ($ S A) \<union> free_tv ($ S t)"
inject_2: fixes x1 :: "name"   and x2 :: "trm"   and y1 :: "name"   and y2 :: "trm" shows "(Lam [x1].x2 = Lam [y1].y2) = ([x1].x2 = [y1].y2)"
assms_2: shows "Group.monoid (G\<lparr>carrier := H\<rparr>)"
card_nlists: fixes A :: "'a set"   and n :: "nat" shows "card (nlists A n) = card A ^ n"
two_composable_identity_arrows_of_spans_axioms: shows "two_composable_identity_arrows_of_spans (\<cdot>) prj0 prj1 \<pi> \<rho>"
span_scale: fixes x :: "'c"   and S :: "'c set"   and c :: "complex" assumes "x \<in> cspan S" shows "c *\<^sub>C x \<in> cspan S"
lmember_def: fixes x :: "'c"   and xs :: "'c llist" shows "lmember x xs = (x \<in> lset xs)"
ODnum_equiv_segment: fixes D :: "'c Order"   and x :: "'c Order set" assumes "Worder D"   and "x = ordinal_number D" shows "ord_equiv (Iod ODnods (segment ODnods x)) D"
simps_2: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and Doc :: "character list"   and k :: "nat"   and u :: "nat" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel" shows "LocalLexing.\<J> \<TT> \<RR> \<SS> Lex Sel Doc k (Suc u) = LocalLexing.\<pi> \<RR> k (LocalLexing.\<T> \<TT> \<RR> \<SS> Lex Sel Doc k (Suc u)) (LocalLexing.\<J> \<TT> \<RR> \<SS> Lex Sel Doc k u)"
Done_eq_DONE: fixes b :: "'d option"   and a :: "'d" shows "(Resumption.resumption.Done b = DONE a) = (b = Some a)"
divide_nonpos_pos: fixes x :: "'a"   and y :: "'a" assumes "x \<le> (0::'a)"   and "(0::'a) < y" shows "x / y \<le> (0::'a)"
assms_2: shows "fmdom' f \<subseteq> fdom"
bb_same: fixes j' :: "nat"   and i' :: "nat"   and j :: "nat"   and i :: "nat"   and k :: "nat" assumes "((j', i'), j, i) \<in> pair_less"   and "(j', i') \<in> IJ k" shows "b k (j', i') \<lless> b k (j, i)"
notret: shows "\<forall>n'. \<pi>' n' \<noteq> local.return"
isTrue_implies_prv_neg_PPf: fixes n :: "'b"   and \<phi> :: "'c" assumes "n \<in> num"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "isTrue (B.neg (PPf n \<langle>\<phi>\<rangle>))" shows "bprv (B.neg (PPf n \<langle>\<phi>\<rangle>))"
pbd_points_index1: fixes ps :: "'a set" assumes "ps \<subseteq> \<V>"   and "card ps = 2" shows "\<B> index ps = 1"
wiring_2: shows "\<I>1,\<I>2 \<turnstile>\<^sub>C conv \<sim> map_converter id id f g 1\<^sub>C"
mr_cos_h_stable: fixes H :: "'a set"   and h :: "'a" assumes "submodule R M H"   and "h \<in> H" shows "H = h \<uplus>\<^bsub>M\<^esub> H"
subpath_to_frontier_explicit: fixes g :: "real \<Rightarrow> 'a"   and S :: "'a set" assumes "path g"   and "pathfinish g \<notin> S" obtains u :: "real" where "0 \<le> u" and "u \<le> 1" and "\<And>x. 0 \<le> x \<and> x < u \<Longrightarrow> g x \<in> interior S" and "g u \<notin> interior S" and "u = 0 \<or> g u \<in> closure S"