hcju/isabelleps huggingface dataset:
Supraclassical_logic_axioms_def: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "Supraclassical_logic_axioms Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL \<equiv> ((((\<forall>A. true_PL \<in> Cn A) \<and> (\<forall>p. p \<in> Cn {false_PL})) \<and> (\<forall>A p q. q \<in> Cn (A \<union> {p}) \<longrightarrow> imp_PL p q \<in> Cn A) \<and> (\<forall>A p q. imp_PL p q \<in> Cn A \<longrightarrow> p \<in> Cn A \<longrightarrow> q \<in> Cn A)) \<and> ((\<forall>A p. imp_PL p false_PL \<in> Cn A \<longrightarrow> not_PL p \<in> Cn A) \<and> (\<forall>A p. not_PL p \<in> Cn A \<longrightarrow> imp_PL p false_PL \<in> Cn A)) \<and> (\<forall>A p q. p \<in> Cn A \<longrightarrow> q \<in> Cn A \<longrightarrow> conj_PL p q \<in> Cn A) \<and> (\<forall>A p q. conj_PL p q \<in> Cn A \<longrightarrow> p \<in> Cn A)) \<and> (((\<forall>A p q. conj_PL p q \<in> Cn A \<longrightarrow> q \<in> Cn A) \<and> (\<forall>A p q. p \<in> Cn A \<longrightarrow> disj_PL p q \<in> Cn A)) \<and> (\<forall>A q p. q \<in> Cn A \<longrightarrow> disj_PL p q \<in> Cn A) \<and> (\<forall>A p q r. disj_PL p q \<in> Cn A \<longrightarrow> imp_PL p r \<in> Cn A \<longrightarrow> imp_PL q r \<in> Cn A \<longrightarrow> r \<in> Cn A)) \<and> ((\<forall>A p q. imp_PL p q \<in> Cn A \<longrightarrow> imp_PL q p \<in> Cn A \<longrightarrow> equiv_PL p q \<in> Cn A) \<and> (\<forall>A p q. equiv_PL p q \<in> Cn A \<longrightarrow> imp_PL p q \<in> Cn A)) \<and> (\<forall>A p q. equiv_PL p q \<in> Cn A \<longrightarrow> imp_PL q p \<in> Cn A) \<and> (\<forall>A p. not_PL (not_PL p) \<in> Cn A \<longrightarrow> p \<in> Cn A) \<and> (\<forall>A p. disj_PL p (not_PL p) \<in> Cn A)"
assms: shows "wiring \<I> \<I>' cnv (f, g)"
zero_less_iff_neq_zero: fixes n :: "'c multiset" shows "({#} \<subset># n) = (n \<noteq> {#})"
add_le_impl: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry"   and c :: "'c DBMEntry" assumes "dbm_add a b \<prec> dbm_add a c" shows "b \<prec> c"
vsv_vdomain_vsingleton_vrange_vsingleton: fixes a :: "V" assumes "\<D>\<^sub>\<circ> F = ZFC_in_HOL.set {a}" obtains b :: "V" where "\<R>\<^sub>\<circ> F = ZFC_in_HOL.set {b}"
assms_1: fixes op :: "char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat" assumes "op \<in> set (ast_problem.ast\<delta> prob)" shows "consistent_pres_op op"
emeasure_density_const: fixes A :: "'a set"   and M :: "'a measure"   and c :: "ennreal" assumes "A \<in> sets M" shows "emeasure (density M (\<lambda>_. c)) A = c * emeasure M A"
inf_continuous_cmul: fixes f :: "'a \<Rightarrow> ennreal"   and c :: "ennreal" assumes "inf_continuous f"   and "c < \<top>" shows "inf_continuous (\<lambda>x. c * f x)"
derivation_ge_list_all: fixes D :: "(nat \<times> symbol \<times> symbol list) list"   and i :: "nat" shows "derivation_ge D i = list_all (\<lambda>d. i \<le> fst d) D"
rtrancl_is_path: fixes u :: "'b"   and v :: "'b"   and E :: "('b \<times> 'b) set" assumes "(u, v) \<in> E\<^sup>*" obtains l :: "'b list" where "path E u l v"
triangle_Suc: fixes n :: "nat" shows "triangle (Suc n) = triangle n + Suc n"
coplanar_perm_4: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "Tarski_neutral_dimensionless.Coplanar Bet A D B C"
subdiv_ends_G_rev_4: shows "head H (rev_G uv) = u"
cut_le_in_imp: fixes x :: "'a"   and I :: "'a set"   and t :: "'a" assumes "x \<in> I \<down>\<le> t" shows "x \<in> I"
elims: fixes x :: "'b list"   and y :: "'b list" assumes "alternate x = y" obtains "x = []" and "y = []"   | xa :: "'b" and xs :: "'b list" where "x = xa # xs" and "y = xa # alternate (tl xs)"
reindex_nontrivial: fixes A :: "'b set"   and h :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'd" assumes "finite A"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; x \<noteq> y; h x = h y\<rbrakk> \<Longrightarrow> g (h x) = (0::'d)" shows "sum g (h ` A) = sum (g \<circ> h) A"
butlast_snoc: fixes xs :: "'e list"   and x :: "'e" shows "butlast (xs @ [x]) = xs"
subspace_Times: fixes s1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and s2 :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" assumes "module_prod s1 s2"   and "module.subspace s1 A"   and "module.subspace s2 B" shows "module.subspace (module_prod.scale s1 s2) (A \<times> B)"
filter_not_empty: fixes x :: "'e"   and xs :: "'e list"   and P :: "'e \<Rightarrow> bool" assumes "x \<in> set xs"   and "P x" shows "filter P xs \<noteq> []"
oneway: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and n :: "nat"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" assumes "\<Turnstile>\<^sub>3\<^sub>' {embP3 P \<and>* $ n} c { embP2 Q}" shows "validk P c Q n"
mem_fsps_snd_mHom: fixes N :: "('g, 'b, 'h) Module_scheme"   and H :: "'a set"   and f :: "'a \<Rightarrow> 'g"   and a :: "'a set"   and b :: "'a \<Rightarrow> 'g" assumes "R module N"   and "free_generator R M H"   and "f \<in> H \<rightarrow> carrier N"   and "(a, b) \<in> fsps R M N f H" shows "b \<in> mHom R (mdl M (fgs R M a)) N"
stop: fixes g :: "('a \<Rightarrow> bool) \<times> ('a \<Rightarrow> 'b \<times> 'a)"   and s :: "'a" assumes "\<not> fst g s" shows "s \<in> terminates_on g"
a_in_ar_coset: fixes R :: "('d, 'e) Ring_scheme"   and I :: "'d set"   and a :: "'d" assumes "Ring R"   and "ideal R I"   and "a \<in> carrier R" shows "a \<in> a \<uplus>\<^bsub>R\<^esub> I"
a: shows "TermEquivCl' T A e B = TermEquivCl' T A d B"
less_le_not_le: fixes x :: "('a, 'b) Term.term"   and y :: "('a, 'b) Term.term" shows "(x <\<cdot> y) = (x \<le>\<cdot> y \<and> \<not> y \<le>\<cdot> x)"
INF_top: fixes U\<^sub>2 :: "'c set"   and f :: "'c \<Rightarrow> 'al"   and A :: "'c set" assumes "\<forall>x\<in>U\<^sub>2. f x = \<top>\<^sub>o\<^sub>w"   and "A \<subseteq> U\<^sub>2" shows "\<Sqinter>\<^sub>o\<^sub>w (f ` A) = \<top>\<^sub>o\<^sub>w"
Labeling_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat" shows "Labeling c s t f l \<equiv> NPreflow c s t f \<and> Labeling_axioms c s t f l"
M_1: fixes m :: "int" assumes "poly_mod_2 m" shows "poly_mod.M m 1 = 1"
edges_add_edge: fixes v :: "'a"   and e :: "'b"   and v' :: "'a"   and g :: "('a, 'b, 'c) graph_scheme" shows "edges (add_edge v e v' g) = insert (v, e, v') (edges g)"
ClassT: fixes statC :: "qtname" assumes "statT = ClassT statC"   and "dynmethd G statC dynC sig = x" shows "thesis"
prj1_simps_2: fixes f :: "'a"   and g :: "'a" assumes "C.cospan f g" shows "C.dom \<p>\<^sub>1[f, g] = f \<down>\<down> g"
simps_3: fixes c :: "'a"   and s :: "'a FOL_Fitting.term"   and i :: "'b"   and l :: "'a FOL_Fitting.term list" shows "subc c s (Pred i l) = Pred i (subc_list c s l)"
SUP_lessD: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and f :: "'d \<Rightarrow> 'b"   and A :: "'d set"   and y :: "'b"   and i :: "'d" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "less (Sup (f ` A)) y"   and "i \<in> A" shows "less (f i) y"
phiUsesE: fixes v :: "'val"   and g :: "'g"   and n :: "'node" assumes "v \<in> step.phiUses g n" obtains n' :: "'node" and v' :: "'val" and vs :: "'val list" where "n' \<in> set (old.successors g n)" and "(n, v) \<in> set (zip (old.predecessors g n') vs)" and "p_g g (n', v') = Some vs"
fsps_chain_boundTr2_1: fixes N :: "('g, 'b, 'h) Module_scheme"   and H :: "'a set"   and f :: "'a \<Rightarrow> 'g"   and C :: "('a set \<times> ('a \<Rightarrow> 'g)) set"   and a :: "'a set"   and b :: "'a \<Rightarrow> 'g"   and aa :: "'a set"   and ba :: "'a \<Rightarrow> 'g"   and x :: "'a" assumes "R module N"   and "free_generator R M H"   and "f \<in> H \<rightarrow> carrier N"   and "C \<subseteq> fsps R M N f H"   and "(a, b) \<in> C"   and "(aa, ba) \<in> C"   and "x \<in> fgs R M a"   and "x \<in> fgs R M aa"   and "a \<subseteq> aa" shows "b x = ba x"
sel_2: shows "ImZ \<i>\<^sub>\<int> = 1"
ae: shows "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. Pxyz x * log b (Pxyz x / (Px (fst x) * Pyz (snd x))) - Pxyz x * log b (Pxz (fst x, snd (snd x)) / (Px (fst x) * Pz (snd (snd x)))) = Pxyz x * log b (Pxyz x * Pz (snd (snd x)) / (Pxz (fst x, snd (snd x)) * Pyz (snd x)))"
map_poly_idI: fixes p :: "'a poly"   and f :: "'a \<Rightarrow> 'a" assumes "\<And>x. x \<in> set (coeffs p) \<Longrightarrow> f x = x" shows "map_poly f p = p"
simps_1: fixes var :: "nat"   and a :: "'a" shows "atom_var (Leq var a) = var"
take_bit_eq_mask_iff: fixes n :: "nat"   and k :: "int" shows "(take_bit n k = mask n) = (take_bit n (k + 1) = 0)"
min_def: fixes a :: "'a Word.word"   and b :: "'a Word.word" shows "signed.min a b = (if a \<le>s b then a else b)"
set_union_code: fixes xs :: "'b list"   and ys :: "'b list" shows "set xs \<union> set ys = set (xs @ ys)"
no_Check_in_def: fixes s :: "event list"   and r :: "room" shows "no_Check_in s r \<equiv> \<nexists>g c. Check_in g r c \<in> set s"
h2: shows "(\<exists>m\<in>monomials p. lookup m var = n) \<and> (\<forall>m\<in>monomials p. lookup m var \<le> n)"
c: shows "f \<approx>>\<^bsub>C\<^esub> g"
poly_uminus_def: fixes p :: "'a poly" shows "poly_uminus p \<equiv> \<Sum>i\<le>degree p. monom ((- (1::'a)) ^ i * coeff p i) i"
Mp_lift_modulus: fixes f :: "int poly"   and g :: "int poly"   and k :: "int" assumes "f =m g" shows "poly_mod.Mp (p * k) (Polynomial.smult k f) = poly_mod.Mp (p * k) (Polynomial.smult k g)"
simps_10: fixes t :: "'b EV.term"   and u :: "'b EV.term"   and v :: "'b EV.term" shows "\<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[t, u, v\<^bold>]\<rbrace> = D.\<alpha>' (\<lbrace>t\<rbrace>, \<lbrace>u\<rbrace>, \<lbrace>v\<rbrace>)"
CFG_SSA_code_def: fixes \<alpha>e :: "'a \<Rightarrow> ('b \<times> 'c \<times> 'b) set"   and \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and Entry :: "'a \<Rightarrow> 'b"   and defs :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and "uses" :: "'a \<Rightarrow> ('b, 'd set) mapping"   and phis :: "'a \<Rightarrow> ('b \<times> 'd, 'd list) mapping" shows "CFG_SSA_code \<alpha>e \<alpha>n invar inEdges' Entry defs uses phis \<equiv> CFG_SSA_ext \<alpha>e \<alpha>n invar inEdges' Entry defs (usesOf \<circ> uses) (\<lambda>g. Mapping.lookup (phis g))"
assms_2: shows "s \<cdot> \<sigma> = s \<cdot> \<tau>"
linearityD_2: fixes E :: "(node fset \<times> command_gadget \<times> node fset) list"   and e :: "node fset \<times> command_gadget \<times> node fset"   and f :: "node fset \<times> command_gadget \<times> node fset" assumes "linearity E"   and "e \<in> set E"   and "f \<in> set E"   and "e \<noteq> f" shows "fst3 e |\<inter>| fst3 f = {||} \<and> thd3 e |\<inter>| thd3 f = {||}"
continuous_on_closure_sequentially': fixes C :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and u :: "nat \<Rightarrow> 'a"   and l :: "'a" assumes "continuous_on (closure C) f"   and "\<And>n. u n \<in> C"   and "u \<longlonglongrightarrow> l" shows "(\<lambda>n. f (u n)) \<longlonglongrightarrow> f l"
less_pos: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and pos :: "'a \<Rightarrow> int"   and x :: "'a"   and y :: "'a" assumes "class.spark_enum less_eq less pos" shows "less x y = (pos x < pos y)"
rcos_sum: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G" shows "derived G (carrier G) #> x <#> (derived G (carrier G) #> y) = derived G (carrier G) #> x \<otimes> y"
power2_eq_iff: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.idom times one plus zero minus uminus" shows "(power.power one times x 2 = power.power one times y 2) = (x = y \<or> x = uminus y)"
cycle_free_to_k: shows "cycle_free_up_to m k n"
perp_col2: fixes A :: "'p"   and B :: "'p"   and X :: "'p"   and Y :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Perp X Y"   and "C \<noteq> D"   and "Col A B C"   and "Col A B D" shows "C D Perp X Y"
real_of_2_Irrational: fixes n :: "nat"   and rai :: "int poly \<times> rat \<times> rat"   and x :: "rat" assumes "invariant_2 (Irrational n rai)" shows "real_of_2 (Irrational n rai) \<noteq> real_of_rat x"
abs_eq: fixes xa :: "perm"   and x :: "'a \<times> 'b" shows "xa \<bullet> abs_residual x = abs_residual (xa \<bullet> x)"
succ_rel_insert_head: fixes ref :: "'b"   and list :: "'b list"   and oid :: "'b" assumes "distinct (ref # list)" shows "succ_rel (insert_spec (ref # list) (oid, Some ref)) = insert_alt (succ_rel (ref # list)) (oid, ref)"
rxa_in_Rxa: fixes R :: "('c, 'd) Ring_scheme"   and a :: "'c"   and r :: "'c" assumes "Ring R"   and "a \<in> carrier R"   and "r \<in> carrier R" shows "r \<cdot>\<^sub>r\<^bsub>R\<^esub> a \<in> R \<diamondsuit>\<^sub>p a"
simps_13: fixes v :: "fm"   and va :: "fm" shows "conjuncts (Or v va) = [Or v va]"
mult_semi_associative: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x * y * z \<le> x * (y * z)"
bias_def: fixes x :: "('a, 'b) IEEE.float itself" shows "bias x = 2 ^ (LENGTH('a) - 1) - 1"
ring_hom_ringI2: fixes R :: "('a, 'b) ring_scheme"   and S :: "('c, 'd) ring_scheme"   and h :: "'a \<Rightarrow> 'c" assumes "ring R"   and "ring S"   and "h \<in> ring_hom R S" shows "ring_hom_ring R S h"
cospoly_lower_nonneg: fixes x :: "real"   and n :: "nat" assumes "0 \<le> x" shows "cospoly (Suc (Suc (Suc (4 * n)))) x \<le> cos x"
intervalPO: fixes a :: "'a"   and b :: "'a" assumes "a \<in> A"   and "b \<in> A"   and "interval r a b \<noteq> {}" shows "\<lparr>pset = interval r a b, order = induced (interval r a b) r\<rparr> \<in> PartialOrder"
assms_5: fixes x :: "'a" assumes "x \<in> S" shows "g (f x) = x"
diagseq_seqseq: fixes P :: "nat \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> bool"   and k :: "nat" assumes "Diagonal_Subsequence.subseqs P" shows "subseqs.diagseq P \<circ> (+) k = subseqs.seqseq P k \<circ> (\<lambda>x. subseqs.fold_reduce P k x (k + x))"
seqE: fixes g :: "'c"   and f :: "'c" assumes "Cop.seq g f" obtains "Cop.arr f" and "Cop.arr g" and "Cop.dom g = Cop.cod f"
semiring_norm_79: fixes m :: "num"   and n :: "num" shows "(num.Bit0 m < num.Bit1 n) = (m \<le> n)"
snd_eqD: fixes x :: "'e"   and y :: "'d"   and a :: "'d" assumes "snd (x, y) = a" shows "y = a"
induct: fixes P :: "Arity list \<Rightarrow> stack_elem list \<Rightarrow> bool"   and a0 :: "Arity list"   and a1 :: "stack_elem list" assumes "\<And>uu_. P uu_ []"   and "\<And>a as e1 e2 S. P as S \<Longrightarrow> P (a # as) (Alts e1 e2 # S)"   and "\<And>as x S. P as S \<Longrightarrow> P as (Arg x # S)"   and "\<And>v va S. P [] S \<Longrightarrow> P [] (Alts v va # S)"   and "\<And>v S. P [] S \<Longrightarrow> P [] (Upd v # S)"   and "\<And>v S. P [] S \<Longrightarrow> P [] (Dummy v # S)"   and "\<And>as v S. P as S \<Longrightarrow> P as (Upd v # S)"   and "\<And>as v S. P as S \<Longrightarrow> P as (Dummy v # S)" shows "P a0 a1"
rep_eq: fixes x :: "String.literal"   and xa :: "String.literal" shows "literal.explode (x + xa) = literal.explode x @ literal.explode xa"
fun_app: fixes V :: "val set"   and m :: "ty"   and n :: "ty"   and \<eta> :: "val set list"   and V' :: "val set" assumes "V \<subseteq> T (m \<rightarrow> n) \<eta>"   and "V' \<subseteq> T m \<eta>" shows "apply_fun V V' \<subseteq> T n \<eta>"
ms_1: shows "DirProds (\<lambda>m. Z (ms ! m)) {..<length ms} \<cong> G"
distinct_2: fixes x21 :: "char list"   and x22 :: "aexp" shows "x21 ::= x22 \<noteq> com.SKIP"
when_cong: fixes P :: "bool"   and Q :: "bool"   and a :: "'c"   and b :: "'c"   and zero :: "'c" assumes "P = Q"   and "Q \<Longrightarrow> a = b" shows "zero.when zero a P = zero.when zero b Q"
induct: fixes P :: "'b tree234 \<Rightarrow> 'b \<Rightarrow> 'b up\<^sub>d \<Rightarrow> 'b \<Rightarrow> 'b tree234 \<Rightarrow> 'b \<Rightarrow> 'b tree234 \<Rightarrow> bool"   and a0 :: "'b tree234"   and a1 :: "'b"   and a2 :: "'b up\<^sub>d"   and a3 :: "'b"   and a4 :: "'b tree234"   and a5 :: "'b"   and a6 :: "'b tree234" assumes "\<And>t1 a t2 b t3 c t4. P t1 a (T\<^sub>d t2) b t3 c t4"   and "\<And>t1 a t2 b t3 c t4 d t5. P \<langle>t1, a, t2\<rangle> b (Up\<^sub>d t3) c t4 d t5"   and "\<And>t1 a t2 b t3 c t4 d t5 e t6. P \<langle>t1, a, t2, b, t3\<rangle> c (Up\<^sub>d t4) d t5 e t6"   and "\<And>t1 a t2 b t3 c t4 d t5 e t6 f t7. P \<langle>t1, a, t2, b, t3, c, t4\<rangle> d (Up\<^sub>d t5) e t6 f t7"   and "\<And>b v d e f g. P \<langle>\<rangle> b (Up\<^sub>d v) d e f g" shows "P a0 a1 a2 a3 a4 a5 a6"
inter_aform_plane_refine: fixes d :: "nat"   and xi :: "(real \<times> real pdevs) list"   and x :: "real list set"   and si :: "real list sctn"   and s :: "real list sctn"   and optns :: "nat" assumes "0 < d"   and "(xi, x) \<in> aforms_rel"   and "(si, s) \<in> \<langle>Id\<rangle>sctn_rel"   and "length xi = d"   and "length (normal si) = d" shows "(nres_of (inter_aform_plane_lv (length xi) optns xi si), inter_sctn_specs d x s) \<in> \<langle>aforms_rel\<rangle>nres_rel"
simps_1: fixes I :: "('b, 'c, 'a) interp"   and x :: "'a" shows "dterm_sem I (Var x) = (\<lambda>v. fst v $ x)"
map_closed: fixes a :: "'a" assumes "a \<in> A" shows "(g \<circ> f \<down> A) a \<in> C"
rep_eq: fixes x :: "'b ratfps"   and xa :: "'b ratfps" shows "Rep_ratfps (x * xa) = Rep_ratfps x * Rep_ratfps xa"
rroots2: fixes p :: "real poly" assumes "degree p = 2" shows "{x. poly p x = 0} = set (rroots2 p)"
lc: assumes "p \<noteq> []" shows "lead_coeff p \<noteq> \<zero>\<^bsub>R\<^esub>"
assms_3: fixes t :: "'b" assumes "t \<in> T" shows "l t \<noteq> b"
simps_2: fixes \<sigma> :: "'e \<Rightarrow> ('c monom \<times> 'd) list"   and m :: "'e monom"   and c :: "'d"   and p :: "('e monom \<times> 'd) list" shows "poly_subst \<sigma> ((m, c) # p) = poly_add (poly_mult [(1, c)] (monom_subst \<sigma> m)) (poly_subst \<sigma> p)"
total_recursive_def1: shows "total_recursive = (\<lambda>f. c_graph f \<in> ce_sets)"
learn_limE_2: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option"   and f :: "nat \<Rightarrow> nat option" assumes "learn_lim \<psi> U s"   and "f \<in> U" shows "\<exists>i. \<psi> i = f \<and> (\<exists>n\<^sub>0. \<forall>n\<ge>n\<^sub>0. s (f \<triangleright> n) \<down>= i)"
hbt_delete_1: fixes m :: "nat"   and t :: "('b \<times> nat) tree"   and x :: "'b" assumes "HBT m"   and "HBT.hbt m t" shows "HBT.hbt m (HBT.delete m x t)"
r_def: fixes A :: "((real, 'a) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec"   and x :: "(real, 'a) Finite_Cartesian_Product.vec" assumes "perron_frobenius A" shows "perron_frobenius.r A x = Min {(A *v x) $h j / x $h j |j. x $h j \<noteq> 0}"
power_mult: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and m :: "nat"   and n :: "nat" assumes "class.monoid_mult one times" shows "power.power one times a (m * n) = power.power one times (power.power one times a m) n"
oracle_decrypt1'_simps: fixes key :: "'prf_key \<times> 'upf_key"   and bad :: "bool"   and L :: "(bool list \<times> bool list \<times> 'hash) set"   and x' :: "bool list"   and c' :: "bool list"   and t' :: "'hash" shows "oracle_decrypt1' key (bad, L) (x', c', t') = return_spmf (None, bad \<or> upf_fun (snd key) (x' @ c') = t' \<and> (x', c', t') \<notin> L \<and> length x' = prf_dlen, L)"
assms_4: shows "is_idirprod (generate G gs) (\<lambda>g. generate G {g}) gs"
hyps_1: fixes t :: "('a, 'b) tm" assumes "x = num_args t - 1"   and "is_App t" shows "t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b chop t"
encL: fixes S :: "'procS"   and TRel :: "('procT \<times> 'procT) set" shows "TargetTerm (\<lbrakk>S\<rbrakk>) \<R>\<lbrakk>\<cdot>\<rbrakk>T<TRel> SourceTerm S"
Sup_trans_def: fixes S :: "(('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real) set" shows "Sup_trans S = (\<lambda>P. Sup_exp {t P |t. t \<in> S})"
healthy_intros_17: fixes S :: "'b \<Rightarrow> 'a set"   and prog :: "'a \<Rightarrow> bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "\<And>x s. x \<in> S s \<Longrightarrow> healthy (wp (prog x))"   and "\<And>s. \<exists>x. x \<in> S s" shows "healthy (wp (SetDC prog S))"
simps_3: fixes M :: "'a"   and N :: "'a"   and P :: "('a, 'b, 'c) psi" shows "extractFrame (M\<langle>N\<rangle>.P) = \<bottom>\<^sub>F"
induct: fixes x :: "event list"   and P :: "event list \<Rightarrow> bool" assumes "x \<in> orb"   and "P []"   and "\<And>evsa X B. \<lbrakk>evsa \<in> orb; P evsa; X \<in> synth (analz (knows Spy evsa))\<rbrakk> \<Longrightarrow> P (Says Spy B X # evsa)"   and "\<And>evsr A B X. \<lbrakk>evsr \<in> orb; P evsr; Says A B X \<in> set evsr\<rbrakk> \<Longrightarrow> P (Gets B X # evsr)"   and "\<And>evs1 NA A B M. \<lbrakk>evs1 \<in> orb; P evs1; Nonce NA \<notin> used evs1\<rbrakk> \<Longrightarrow> P (Says A B \<lbrace>Nonce M, Agent A, Agent B, Crypt (shrK A) \<lbrace>Nonce NA, Nonce M, Agent A, Agent B\<rbrace>\<rbrace> # evs1)"   and "\<And>evs2 NB B M A X. \<lbrakk>evs2 \<in> orb; P evs2; Nonce NB \<notin> used evs2; Gets B \<lbrace>Nonce M, Agent A, Agent B, X\<rbrace> \<in> set evs2\<rbrakk> \<Longrightarrow> P (Says B Server \<lbrace>Nonce M, Agent A, Agent B, X, Crypt (shrK B) \<lbrace>Nonce NB, Nonce M, Nonce M, Agent A, Agent B\<rbrace>\<rbrace> # evs2)"   and "\<And>evs3 KAB M A B NA NB. \<lbrakk>evs3 \<in> orb; P evs3; Key KAB \<notin> used evs3; Gets Server \<lbrace>Nonce M, Agent A, Agent B, Crypt (shrK A) \<lbrace>Nonce NA, Nonce M, Agent A, Agent B\<rbrace>, Crypt (shrK B) \<lbrace>Nonce NB, Nonce M, Nonce M, Agent A, Agent B\<rbrace>\<rbrace> \<in> set evs3\<rbrakk> \<Longrightarrow> P (Says Server B \<lbrace>Nonce M, Crypt (shrK B) \<lbrace>Crypt (shrK A) \<lbrace>Nonce NA, Key KAB\<rbrace>, Nonce NB, Key KAB\<rbrace>\<rbrace> # evs3)"   and "\<And>evs4 B X M A X' NB KAB. \<lbrakk>evs4 \<in> orb; P evs4; B \<noteq> Server; \<forall>p q. X \<noteq> \<lbrace>p, q\<rbrace>; Says B Server \<lbrace>Nonce M, Agent A, Agent B, X', Crypt (shrK B) \<lbrace>Nonce NB, Nonce M, Nonce M, Agent A, Agent B\<rbrace>\<rbrace> \<in> set evs4; Gets B \<lbrace>Nonce M, Crypt (shrK B) \<lbrace>X, Nonce NB, Key KAB\<rbrace>\<rbrace> \<in> set evs4\<rbrakk> \<Longrightarrow> P (Says B A \<lbrace>Nonce M, X\<rbrace> # evs4)"   and "\<And>evso B M A NA KAB NB. \<lbrakk>evso \<in> orb; P evso; Says Server B \<lbrace>Nonce M, Crypt (shrK B) \<lbrace>Crypt (shrK A) \<lbrace>Nonce NA, Key KAB\<rbrace>, Nonce NB, Key KAB\<rbrace>\<rbrace> \<in> set evso\<rbrakk> \<Longrightarrow> P (Notes Spy \<lbrace>Agent A, Agent B, Nonce NA, Nonce NB, Key KAB\<rbrace> # evso)" shows "P x"
trss_drop_all_context_s: fixes s :: "'f list"   and c :: "'f list multiset"   and e :: "('g, 'h) label"   and s' :: "'f list"   and c' :: "'f list multiset"   and fg :: "('f, 'g, 'h, 'i, 'j) flowgraph_rec_scheme" assumes "((s, c), e, s', c') \<in> trss fg" obtains csp :: "'f list multiset" where "c' = csp + c" and "((s, {#}), e, s', csp) \<in> trss fg"
wf_query_def: fixes n :: "nat"   and V :: "nat set"   and Q_pos :: "(nat set \<times> 'b option list set) set"   and Q_neg :: "(nat set \<times> 'b option list set) set" shows "wf_query n V Q_pos Q_neg = ((\<forall>X\<in>Q_pos \<union> Q_neg. wf_atable n X) \<and> wf_set n V \<and> 1 \<le> card Q_pos)"