hcju/isabelleps huggingface dataset:
inducts: fixes xa :: "real \<times> real"   and x :: "(real \<times> real) list"   and P :: "(real \<times> real) list \<Rightarrow> bool" assumes "ccw'.sortedP xa x"   and "P []"   and "\<And>ys x. \<lbrakk>\<forall>y\<in>set ys. ccw' xa x y; ccw'.sortedP xa ys; P ys\<rbrakk> \<Longrightarrow> P (x # ys)" shows "P x"
wf_mstate_msteps_stateless: fixes \<pi> :: "(String.literal \<times> event_data list) prefix"   and R :: "event_data list set"   and st :: "('a, 'b) mstate"   and v :: "event_data option list"   and \<pi>' :: "(String.literal \<times> event_data list) prefix"   and i :: "nat" assumes "wf_mstate \<phi> \<pi> R st"   and "mem_restr R v"   and "\<pi> \<le> \<pi>'" shows "((i, v) \<in> flatten_verdicts (msteps_stateless (pdrop (plen \<pi>) \<pi>') st)) = ((i, v) \<in> M \<pi>' - M \<pi>)"
vintersection_vunion_left: fixes A :: "V"   and B :: "V"   and C :: "V" shows "(A \<union>\<^sub>\<circ> B) \<inter>\<^sub>\<circ> C = A \<inter>\<^sub>\<circ> C \<union>\<^sub>\<circ> B \<inter>\<^sub>\<circ> C"
to_vec_one: shows "to_vec 1 = mat (1::'c)"
simps: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "at_begin_fst_awtn (as, lm) (s, l, r) ires = (\<exists>lm1 tn rn. lm1 = lm @ 0 \<up> tn \<and> length lm1 = s \<and> (if lm1 = [] then l = Bk # Bk # ires else l = [Bk] @ <rev lm1> @ Bk # Bk # ires) \<and> r = [Oc] @ Bk \<up> rn)"
prime_divisor: fixes j :: "nat"   and n :: "nat" assumes "2 \<le> j"   and "j dvd n" shows "\<exists>p. prime p \<and> p dvd j \<and> p dvd n"
idem: fixes a :: "'a Word.word" shows "signed.max a a = a"
powers_4: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a * b * c = a * (b * c)"
compl_le_swap1: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and y :: "'c"   and x :: "'c" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top"   and "less_eq y (uminus x)" shows "less_eq x (uminus y)"
dijkstra_def: shows "dijkstra \<equiv> dinit \<bind> (\<lambda>\<sigma>0. WHILE\<^sub>T\<^bsup>dinvar\<^esup> (\<lambda>(wl, uu_). wl \<noteq> {}) (\<lambda>\<sigma>. pop_min \<sigma> \<bind> (\<lambda>(x, y). update x y)) \<sigma>0 \<bind> (\<lambda>(uu_, res). RETURN res))"
normal: fixes x :: "('e, 'f) IEEE.float" assumes "is_normal x"   and "valof x = real_of_int (normal_mantissa x) * 2 powr real_of_int (normal_exponent x)" shows "P x"
residue_def: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex" shows "residue f z = (SOME int. \<exists>e>0. \<forall>\<epsilon>>0. \<epsilon> < e \<longrightarrow> (f has_contour_integral complex_of_real (2 * pi) * \<i> * int) (circlepath z \<epsilon>))"
h1: shows "p = (\<Sum>i\<le>MPoly_Type.degree p x. isolate_variable_sparse p x i * Var x ^ i)"
bij_betw_the_inv_into: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set" assumes "bij_betw f A B" shows "bij_betw (the_inv_into A f) B A"
assms_4: shows "path_conforms_with_strategy p P well_ordered_strategy"
sorted_wrt_less_idx: fixes ns :: "nat list"   and i :: "nat" assumes "sorted_wrt (<) ns"   and "i < length ns" shows "i \<le> ns ! i"
assertion_axioms: shows "assertion (\<otimes>) SImp' (\<one>) SChanEq'"
SKIP: fixes G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set" shows "G \<rhd> Skip : Sec (\<lambda>(s, t).  s \<approx> t )"
list_conj_def: fixes ps :: "fm list" shows "list_conj ps \<equiv> foldr MIR.conj ps T"
sup_continuous_compose: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'a" assumes "sup_continuous f"   and "sup_continuous g" shows "sup_continuous (\<lambda>x. f (g x))"
PRelQ: fixes y :: "name" assumes "y \<in> supp (P, Q, x)" shows "(P[x::=y], Q[x::=y]) \<in> Rel"
diagonalize_strict_arr: fixes f :: "'c local.term set" assumes "arr f" shows "\<F>C.D f = f"
subset_h: shows "consistent_sign_vectors ps UNIV \<subseteq> set (map (undo_factorize_polys data) sgas)"
freedb_closing_subst: fixes e :: "db"   and k :: "nat"   and \<Gamma> :: "nat \<Rightarrow> db"   and i :: "nat" assumes "\<forall>v. freedb e v \<and> k \<le> v \<longrightarrow> closed (\<Gamma> (v - k))" shows "freedb (closing_subst e \<Gamma> k) i = (freedb e i \<and> i < k)"
simps_162: fixes x21 :: "('b, 'a, 'c) expr"   and x22 :: "('b, 'a, 'c) expr" shows "LID\<^sub>E (Apply x21 x22) = LID\<^sub>E x21 \<union> LID\<^sub>E x22"
holomorphic_factor_order_of_zero_strong: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and \<xi> :: "complex"   and n :: "nat" assumes "f holomorphic_on S"   and "open S"   and "\<xi> \<in> S"   and "0 < n"   and "(deriv ^^ n) f \<xi> \<noteq> 0"   and "\<And>i. \<lbrakk>0 < i; i < n\<rbrakk> \<Longrightarrow> (deriv ^^ i) f \<xi> = 0" obtains g :: "complex \<Rightarrow> complex" and r :: "real" where "0 < r" and "g holomorphic_on ball \<xi> r" and "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> f w - f \<xi> = ((w - \<xi>) * g w) ^ n" and "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> g w \<noteq> 0"
pseudofunctor_pseudonaturally_equivalent_to_equivalence_pseudofunctor_def: fixes V\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>C :: "'c \<Rightarrow> 'c"   and src\<^sub>C :: "'c \<Rightarrow> 'c"   and trg\<^sub>C :: "'c \<Rightarrow> 'c"   and V\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>D :: "'d \<Rightarrow> 'd"   and src\<^sub>D :: "'d \<Rightarrow> 'd"   and trg\<^sub>D :: "'d \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>F :: "'c \<times> 'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>G :: "'c \<times> 'c \<Rightarrow> 'd"   and \<tau>\<^sub>0 :: "'c \<Rightarrow> 'd"   and \<tau>\<^sub>1 :: "'c \<Rightarrow> 'd" shows "pseudofunctor_pseudonaturally_equivalent_to_equivalence_pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1 \<equiv> (bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C \<and> bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D) \<and> equivalence_pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F \<and> pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G \<and> pseudonatural_equivalence V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1"
Oops1: fixes evsO1 :: "event list"   and A :: "agent"   and authK :: "nat"   and Ta :: "nat"   and authTicket :: "msg" assumes "evsO1 \<in> kerbIV_gets"   and "A \<noteq> Spy"   and "Says Kas A (Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta, authTicket\<rbrace>) \<in> set evsO1"   and "expiredAK Ta evsO1" shows "Says A Spy \<lbrace>Agent A, Agent Tgs, Number Ta, Key authK\<rbrace> # evsO1 \<in> kerbIV_gets"
symcl_def: fixes R :: "('e \<times> 'e) set" shows "R\<^sup>s = R \<union> (\<lambda>(a, b). (b, a)) ` R"
sp_def: shows "sp \<equiv> \<lambda>n. \<Prod>j. d (j + n)"
qfN_zero: fixes N :: "int"   and a :: "int"   and b :: "int" assumes "1 \<le> N"   and "a\<^sup>2 + N * b\<^sup>2 = 0" shows "a = 0 \<and> b = 0"
mul_instr_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = snd (fst (mul_instr instr s1)) \<and> t2 = snd (fst (mul_instr instr s2))" shows "low_equal t1 t2"
normalize_coprime: fixes r :: "int \<times> int"   and p :: "int"   and q :: "int" assumes "Rat.normalize r = (p, q)" shows "coprime p q"
simps_3: fixes x :: "('a, 'b) Combinators"   and v :: "('a, 'b) Combinators"   and va :: "('a, 'b) Combinators list" shows "list2FWpolicy (x # v # va) = x \<oplus> list2FWpolicy (v # va)"
opar_comp_def: fixes s :: "((nat \<Rightarrow> 'e) \<times> 'f, 'h seq_action) automaton"   and i :: "nat"   and t :: "('g, 'h seq_action) automaton" shows "s \<langle>\<langle>\<^bsub>i\<^esub> t \<equiv> \<lparr>init = extg ` (init s \<times> init t), trans = oparp_sos i (automaton.trans s) (automaton.trans t)\<rparr>"
SV_inv3_inj_on_fst_vote_set: fixes s :: "'b v_state_scheme"   and Q :: "process set" assumes "s \<in> SV_inv3" shows "inj_on fst (vote_set (votes s) Q)"
P2: shows "A'' \<noteq> B'' \<and> C'' \<noteq> B''"
delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M_ptr_in_heap: fixes h :: "(_) heap"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and h' :: "(_) heap" assumes "h \<turnstile> delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr \<rightarrow>\<^sub>h h'" shows "shadow_root_ptr |\<in>| shadow_root_ptr_kinds h"
deriv_i_def: fixes d :: "nat \<times> nat \<times> nat" shows "deriv_i d = snd (snd d)"
rangeSetOfPairs: fixes X :: "('d \<times> 'c set) set" shows "Range {(fst pair, Y) |Y pair. Y \<in> finestpart (snd pair) \<and> pair \<in> X} = {Y. \<exists>x. Y \<in> finestpart x \<and> x \<in> Range X}"
PF: shows "PreFunctor (HomP\<^bsub>C\<^esub>[X,\<midarrow>])"
enforce_spmf_alt_def: fixes P :: "'c \<Rightarrow> bool"   and p :: "'c spmf" shows "enforce_spmf P p = p \<bind> (\<lambda>a. assert_spmf (P a) \<bind> (\<lambda>_. return_spmf a))"
ide_dom: fixes f :: "'a \<times> 'a" assumes "VoV.arr f" shows "VoV.ide (VoV.dom f)"
has_integral_interior: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and y :: "'d" assumes "negligible (frontier S)" shows "(f has_integral y) (interior S) = (f has_integral y) S"
wcode_backto_standard_pos_2_via_right: fixes ires :: "cell list"   and rs :: "nat"   and b :: "cell list"   and list :: "cell list" assumes "wcode_goon_right_moving_2 ires rs (b, Bk # list)" shows "wcode_backto_standard_pos_2 ires rs (b, Oc # list)"
simps_4: fixes f1 :: "'a"   and f2 :: "'a" shows "(case Neg of Pos \<Rightarrow> f1 | Neg \<Rightarrow> f2) = f2"
primerec_rec_not_1: shows "primerec rec_not (Suc 0)"
hyps_1: shows " t \<longrightarrow>\<^sub>1\<^sup>* s1"
hyps_1: shows "P \<turnstile> \<langle>e,(h, l(V := None))\<rangle> \<rightarrow> \<langle>e',(h', l')\<rangle>"
Let_2: fixes r :: "C"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue" shows "(\<N>\<lbrakk> e \<rbrakk>\<^bsub>\<rho>\<^esub>)|\<^bsub>C\<cdot>r\<^esub> = (\<N>\<lbrakk> e \<rbrakk>\<^bsub>\<rho>|\<^sup>\<circ>\<^bsub>r\<^esub>\<^esub>)|\<^bsub>C\<cdot>r\<^esub>"
5: shows "project visible (s\<^sub>1 @ s\<^sub>2) = project visible (s\<^sub>1 @ s\<^sub>3)"
add_closed: fixes a :: "'m"   and b :: "'m" assumes "a \<in> U\<^sub>M"   and "b \<in> U\<^sub>M" shows "a +\<^sub>M b \<in> U\<^sub>M"
wt_Atm: fixes atm :: "'atom"   and s :: "'state"   and i :: "nat" shows "wt (Atm atm) s i = 1"
mult_isor: fixes x :: "'a ppath set"   and y :: "'a ppath set"   and z :: "'a ppath set" assumes "x \<subseteq> y" shows "pp_prod x z \<subseteq> pp_prod y z"
comp_ide_self: fixes a :: "'c \<times> 'c" assumes "C.VV.ide a" shows "C.VV.comp a a = a"
lfilled_single: fixes k :: "nat"   and lholed :: "Lholed"   and es :: "e list"   and e :: "e" assumes "Lfilled k lholed es [e]"   and "\<And>a b c. e \<noteq> Label a b c" shows "es = [e] \<and> lholed = LBase [] [] \<or> es = []"
ide_in_hom: fixes a :: "'a1" shows "C1.ide a = \<guillemotleft>a : a \<rightarrow>\<^sub>1 a\<guillemotright>"
subst_atm_mset_list_id_subst: fixes AAs :: "'a multiset list" shows "AAs \<cdot>aml id_subst = AAs"
differentiable_onD: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and x :: "'c" assumes "f differentiable_on S"   and "x \<in> S" shows "f differentiable at x within S"
hom_prod_list: fixes hom :: "'b \<Rightarrow> 'c"   and xs :: "'b list" assumes "monoid_mult_hom hom" shows "hom (prod_list xs) = prod_list (map hom xs)"
simps_5: fixes c1 :: "('g, 'h, 'i, 'j) com"   and c2 :: "('g, 'h, 'i, 'j) com"   and aft :: "'h set" shows "fragments (c1 ;; c2) aft = fragments c1 (atC c2) \<union> fragments c2 aft"
sorted_filter: fixes xs :: "'c list"   and P :: "'c \<Rightarrow> bool" assumes "sorted xs" shows "sorted (List.filter P xs)"
scalar_product_def: fixes v :: "('e, 'f) Finite_Cartesian_Product.vec"   and w :: "('e, 'f) Finite_Cartesian_Product.vec" shows "scalar_product v w = (\<Sum>i\<in>UNIV. v $h i * w $h i)"
simps_6: fixes n :: "nat"   and f :: "recf" shows "wellf (Mn n f) = (0 < n \<and> arity f = Suc n \<and> wellf f)"
\<pi>_2: fixes x :: "'b" assumes "x \<in> agents" shows "weak_ranking (R2 (\<pi> x)) = weak_ranking (R1 x)"
eucl_ivals_5: fixes a :: "'b"   and b :: "'b" shows "{a..b} \<in> sets borel"
dom_cod: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C"   and "partial_magma.arr C f" shows "partial_magma.dom C (partial_magma.cod C f) = partial_magma.cod C f"
rel_cong: fixes x :: "'g list"   and ya :: "'g list"   and y :: "'h list"   and xa :: "'h list"   and R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Ra :: "'g \<Rightarrow> 'h \<Rightarrow> bool" assumes "x = ya"   and "y = xa"   and "\<And>z yb. \<lbrakk>z \<in> set ya; yb \<in> set xa\<rbrakk> \<Longrightarrow> R z yb = Ra z yb" shows "list_all2 R x y = list_all2 Ra ya xa"
rGamma_series_Weierstrass_nonpos_Ints: fixes n :: "nat" shows "\<forall>\<^sub>F k in sequentially. rGamma_series_Weierstrass (- of_nat n) k = (0::'a)"
card_of_ordLeq: fixes A :: "'a set"   and B :: "'b set" shows "(\<exists>f. inj_on f A \<and> f ` A \<subseteq> B) = (|A| \<le>o |B|)"
next_action_after_cswitch: shows "\<forall>s n d aseqs. fst (CISK_control (cswitch n s) d aseqs) = fst (CISK_control s d aseqs)"
antisym_conv: fixes y :: "'c list"   and x :: "'c list" assumes "sublist y x" shows "sublist x y = (x = y)"
parCasesB: fixes P :: "pi"   and Q :: "pi"   and a :: "name"   and x :: "name"   and PQ' :: "pi"   and F :: "pi \<Rightarrow> bool" assumes "P \<parallel> Q \<longmapsto> a<\<nu>x> \<prec> PQ'"   and "\<And>P'. \<lbrakk>P \<longmapsto> a<\<nu>x> \<prec> P'; x \<sharp> Q\<rbrakk> \<Longrightarrow> F (P' \<parallel> Q)"   and "\<And>Q'. \<lbrakk>Q \<longmapsto> a<\<nu>x> \<prec> Q'; x \<sharp> P\<rbrakk> \<Longrightarrow> F (P \<parallel> Q')" shows "F PQ'"
rs: fixes r :: "complex poly"   and c :: "complex" assumes "(r, c) \<in> set ts" shows "rs c = r"
degree_Poly': fixes xs :: "'a list" shows "degree (Poly xs) \<le> length xs - 1"
intvs_incr_h: fixes l :: "int"   and h :: "int" assumes "l \<le> h" shows "{l..<h + 1} = insert h {l..<h}"
e: shows "steps (s, l, r) (ci ly (start_of ly as) ins, start_of ly as - 1) stp = (s', l', r')"
limit_cone_axioms: shows "GE.limit_cone (G a) (G \<circ> \<chi>)"
red_single_lookup: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and f :: "'t \<Rightarrow>\<^sub>0 'c"   and t :: "'a" assumes "red_single p q f t" shows "lookup q (t \<oplus> lt f) = (0::'c)"
assms_1: shows "\<sigma> = as @ [x] @ bs @ [x] @ cs"
tt_tail: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "tail p \<noteq> 0" shows "tt (tail p) = tt p"
inj_idom_hom_axioms: shows "inj_idom_hom hom"
spmf_le_weight: fixes p :: "'c spmf"   and x :: "'c" shows "spmf p x \<le> weight_spmf p"
iso_verts_inv_iso: fixes hom :: "('c, 'd, 'e, 'f) digraph_isomorphism"   and u :: "'c" assumes "iG.digraph_isomorphism hom"   and "u \<in> verts (app_iso hom G)" shows "iso_verts (iG.inv_iso hom) (iso_verts hom u) = u"
pseudonatural_equivalence_axioms: shows "pseudonatural_equivalence (\<cdot>\<^sub>B) (\<star>\<^sub>B) \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1"
edka5_tabulate_def: fixes am :: "nat \<Rightarrow> nat list" shows "edka5_tabulate am \<equiv> init_cf \<bind> (\<lambda>cf. init_ps am \<bind> (\<lambda>am. return (cf, am)))"
wf_not_refl: fixes r :: "('c \<times> 'c) set"   and a :: "'c" assumes "wf r" shows "(a, a) \<notin> r"
le_TC_def: fixes x :: "V"   and y :: "V" shows "x \<sqsubseteq> y \<equiv> x \<sqsubset> y \<or> x = y"
simps: fixes A :: "('c, 'd) ngba"   and a1 :: "('c \<times> 'd) list"   and a2 :: "'d" shows "NGBA.path A a1 a2 = ((\<exists>p. a1 = [] \<and> a2 = p) \<or> (\<exists>a p r. a1 = a # r \<and> a2 = p \<and> (fst a \<in> alphabet A \<and> snd a \<in> transition A (fst a) p) \<and> NGBA.path A r (snd a)))"
eqlen: shows "length (c # V) = length (c' # V')"
vec_eq_iff: fixes x :: "('b, 'c) Finite_Cartesian_Product.vec"   and y :: "('b, 'c) Finite_Cartesian_Product.vec" shows "(x = y) = (\<forall>i. x $h i = y $h i)"
inf_commute: fixes x :: "'d"   and y :: "'d" shows "inf x y = inf y x"
ucast_0: fixes w :: "word32" assumes "get_S w = 0" shows "get_S w = 0"
gT_1: shows "\<Gamma> (Fun g T) = Fun g (map \<Gamma> U)"
Nil_is_rev_conv: fixes xs :: "'d list" shows "([] = rev xs) = (xs = [])"
Nml_HcompNml_2: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Nml t"   and "Nml u"   and "Src t = Trg u" shows "Dom (t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u) = Dom t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> Dom u"
symmetric_Q: fixes I :: "'c"   and a :: "'b"   and b :: "'b" assumes "Q I a b" shows "Q I b a"
inv_inv_eq: fixes f :: "'a \<Rightarrow> 'b" assumes "bij f" shows "inv' (inv' f) = f"
simps_4: fixes c_sinvar :: "'c graph \<Rightarrow> bool"   and c_offending_flows :: "'c graph \<Rightarrow> ('c \<times> 'c) set set"   and c_isIFS :: "bool"   and more :: "'b" shows "SecurityInvariant_configured.more \<lparr>c_sinvar = c_sinvar, c_offending_flows = c_offending_flows, c_isIFS = c_isIFS, \<dots> = more\<rparr> = more"
simps_2: fixes R :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a"   and f :: "bool \<Rightarrow> 'a sorter" shows "count_queries R (Query a b f) = Suc (count_queries R (f ((a, b) \<in> R)))"
sensors_def: fixes sensors :: "cars \<Rightarrow> traffic \<Rightarrow> cars \<Rightarrow> real" shows "sensors sensors \<equiv> \<forall>e ts c. 0 < sensors e ts c"
security_and_P1': fixes a1 :: "bool"   and a2 :: "bool"   and b1 :: "bool"   and b2 :: "bool"   and D :: "((bool \<times> bool) \<times> bool \<times> 'v_14_OT1) \<times> bool \<times> bool \<Rightarrow> bool spmf" shows "\<bar>spmf (and_R1 (a1, a2) (b1, b2) \<bind> D) True - spmf (and_funct (a1, a2) (b1, b2) \<bind> (\<lambda>(s1, s2). S1_and' (a1, a2) (b1, b2) s1 \<bind> D)) True\<bar> \<le> adv_14_OT"
diff_submanifold2: fixes k :: "enat"   and charts1 :: "('c, 'd) chart set"   and charts2 :: "('e, 'f) chart set"   and f :: "'c \<Rightarrow> 'e"   and S :: "'e set" assumes "diff k charts1 charts2 f"   and "open S"   and "f ` manifold.carrier charts1 \<subseteq> S" shows "diff k charts1 (manifold.charts_submanifold charts2 S) f"
cases: fixes n :: "'node"   and p :: "'inPort"   and a :: "'outPort" assumes "hyps_for'p n p a" obtains h :: "'outPort" where "a = h" and "hyps n h = Some p"
elims: fixes x :: "'c \<Rightarrow> 'b"   and xa :: "'c list"   and xb :: "'b list"   and y :: "'b list" assumes "map_tailrec_rev x xa xb = y" obtains f :: "'c \<Rightarrow> 'b" and bs :: "'b list" where "x = fa__" and "xa = []" and "xb = bs__" and "y = bs__"   | f :: "'c \<Rightarrow> 'b" and a :: "'c" and as :: "'c list" and bs :: "'b list" where "x = f" and "xa = a # as" and "xb = bs" and "y = map_tailrec_rev f as (f a # bs)"
rel_flip: fixes R :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a list"   and b :: "'b list" shows "list_all2 R\<inverse>\<inverse> a b = list_all2 R b a"
knows_Spy_partsEs_2: fixes K :: "nat"   and X :: "msg"   and H :: "msg set" assumes "Crypt K X \<in> parts H" obtains "X \<in> parts H"
convex_affine_rel_frontier_Int: fixes S :: "'c set"   and T :: "'c set" assumes "convex S"   and "affine T"   and "interior S \<inter> T \<noteq> {}" shows "rel_frontier (S \<inter> T) = frontier S \<inter> T"
addrow_mat_def: fixes n :: "nat"   and a :: "'c"   and k :: "nat"   and l :: "nat" shows "addrow_mat n a k l = mat n n (\<lambda>(i, j). (if k = i \<and> l = j then (+) a else id) (if i = j then 1::'c else (0::'c)))"
rel_def: fixes Rv :: "('V \<times> 'a) set"   and Rw :: "('W \<times> 'b) set" shows "\<langle>Rv, Rw\<rangle>g.rel \<equiv> br g.\<alpha> g.invar O \<langle>Rv, Rw\<rangle>graph_rel"
no_head_gen_code': fixes \<C> :: "'b list set"   and b :: "'b list"   and xs :: "'b list list"   and ys :: "'b list list" assumes "code \<C>"   and "b \<in> \<C>"   and "xs \<in> lists (no_head_gen \<C> b)"   and "ys \<in> lists (no_head_gen \<C> b)"   and "concat xs = concat ys" shows "xs = ys"
span_inv: fixes e :: "'a"   and v :: "'a"   and t :: "'a"   and g :: "'a"   and r :: "'a" assumes "e \<le> v \<sqinter> - id v"   and "v \<sqinter> top = v"   and "(e \<sqinter> top \<sqinter> id (e \<sqinter> top) \<le> top \<and> top \<sqinter> (e \<sqinter> top) = top) \<and> id e \<sqinter> top \<sqinter> id (id e \<sqinter> top) \<le> top \<and> top \<sqinter> (id e \<sqinter> top) = top"   and "t \<le> v \<sqinter> id v \<sqinter> g"   and "local.comp_inf.symmetric g"   and "id v = id r \<sqinter> top"   and "r \<sqinter> id r \<le> top"   and "id r \<le> id v"   and "id r \<sqinter> top \<le> id r \<sqinter> top" shows "id r \<sqinter> top \<le> id r \<sqinter> top"
simps_2: fixes x2 :: "'b"   and y2 :: "'b" shows "(Inr x2 = Inr y2) = (x2 = y2)"
take_in_aforms_errI: fixes xs :: "real list"   and e :: "nat \<Rightarrow> real"   and XS :: "((real \<times> real pdevs) \<times> real) list"   and n :: "nat" assumes "xs \<in> aforms_err e XS" shows "take n xs \<in> aforms_err e (take n XS)"
reduced_calculus_axioms: shows "reduced_calculus Bot Inf (\<Turnstile>) Red_I Red_F"
zabs_less_one_iff: fixes z :: "int" shows "(\<bar>z\<bar> < 1) = (z = 0)"
simps_1: fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and c :: "('a, 'c, 'd) Stmt" assumes "sifum_lang_no_dma eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars" shows "sifum_lang_no_dma.cxt_to_stmt [] c = c"
weaken_right: fixes H :: "'b pl set"   and q :: "'b pl"   and p :: "'b pl" assumes "H |- q" shows "H |- p -> q"
ball_Un: fixes A :: "'k set"   and B :: "'k set"   and P :: "'k \<Rightarrow> bool" shows "(\<forall>x\<in>A \<union> B. P x) = ((\<forall>x\<in>A. P x) \<and> (\<forall>x\<in>B. P x))"
flattening_lemma_i: fixes \<pi> :: "'c strips_operator list list"   and \<Pi> :: "'c strips_problem" assumes "\<forall>ops\<in>set \<pi>. \<forall>op\<in>set ops. op \<in> set (\<Pi>\<^sub>\<O>)" shows "\<forall>op\<in>set (concat \<pi>). op \<in> set (\<Pi>\<^sub>\<O>)"
lappend_lprefixE: fixes xs :: "'c llist"   and ys :: "'c llist"   and zs :: "'c llist" assumes "lappend xs ys \<sqsubseteq> zs" obtains zs' :: "'c llist" where "zs = lappend xs zs'"
properdiv_set: fixes m :: "nat" shows "properdiv_set m = {n. n properdiv m}"
gpv_coinduct_bind: fixes R :: "('a, 'b, 'c) gpv \<Rightarrow> ('a, 'b, 'c) gpv \<Rightarrow> bool"   and gpv :: "('a, 'b, 'c) gpv"   and gpv' :: "('a, 'b, 'c) gpv" assumes "R gpv gpv'"   and "\<And>gpv gpv'. R gpv gpv' \<Longrightarrow> rel_spmf (rel_generat (=) (=) (rel_fun (=) (\<lambda>gpv gpv'. R gpv gpv' \<or> gpv = gpv' \<or> (\<exists>gpv2 gpv2' f f'. gpv = bind_gpv gpv2 f \<and> gpv' = bind_gpv gpv2' f' \<and> rel_gpv (\<lambda>x y. R (f x) (f' y)) (=) gpv2 gpv2')))) (the_gpv gpv) (the_gpv gpv')" shows "gpv = gpv'"