hcju/isabelleps huggingface dataset:
goodE: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'c" assumes "good P f" obtains i :: "nat" and j :: "nat" where "i < j" and "P (f i) (f j)"
subst_set_def: fixes \<sigma> :: "'d \<Rightarrow> ('c, 'd) Term.term"   and E :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) set" shows "subst_set \<sigma> E = (\<lambda>e. (fst e \<cdot> \<sigma>, snd e \<cdot> \<sigma>)) ` E"
set_plus_intro: fixes a :: "'c"   and C :: "'c set"   and b :: "'c"   and D :: "'c set" assumes "a \<in> C"   and "b \<in> D" shows "a + b \<in> C + D"
ipurge_tr_rev_all: fixes D :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and u :: "'a"   and I :: "('a \<times> 'a) set" assumes "\<forall>v\<in>D ` set xs. (v, u) \<in> I" shows "ipurge_tr_rev I D u xs = xs"
u'_def: fixes x :: "'a" shows "u' x = (if x \<in> I then u x else f x)"
map_append: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list"   and ys :: "'c list" shows "map f (xs @ ys) = map f xs @ map f ys"
prems_1: shows "(l, m) \<in> (succsr (dioph_succs n ks))\<^sup>*"
simps_2: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cmp :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and pin :: "'b \<Rightarrow> 'a list set"   and pout :: "'b \<Rightarrow> 'a list"   and bc :: "'b \<Rightarrow> 'a list"   and mining :: "'b \<Rightarrow> bool"   and honest :: "'a \<Rightarrow> bool"   and cb :: "nat"   and n\<^sub>s :: "nat"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'a" assumes "Blockchain active cmp pin pout bc mining honest cb"   and "\<not> (\<exists>n'<n\<^sub>s. \<not> Option.is_none (Blockchain.devBC active cmp bc mining t n nid n'))"   and "Option.is_none (Blockchain.devBC active cmp bc mining t n nid n\<^sub>s)" shows "Blockchain.devExt active cmp bc mining t n nid n\<^sub>s 0 = []"
BuildSeq_combine: fixes B :: "hf \<Rightarrow> bool"   and C :: "hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> bool"   and s1 :: "hf"   and k1 :: "hf"   and y1 :: "hf"   and s2 :: "hf"   and k2 :: "hf"   and y2 :: "hf"   and y :: "hf" assumes "BuildSeq B C s1 k1 y1"   and "BuildSeq B C s2 k2 y2"   and "C y y1 y2" shows "BuildSeq B C (insf (seq_append (succ k1) s1 s2) (succ (succ (k1 @+ k2))) y) (succ (succ (k1 @+ k2))) y"
fail_butlast_simp_1: shows "butlast (io_init @ [io_last] || tr2 @ [fst FAIL]) = io_init || tr2"
unifiable_imp_empty_or_unif: fixes E :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) multiset" assumes "unifiable (set_mset E)" shows "E = {#} \<or> (\<exists>E'. (E, E') \<in> unif)"
subst_inj_is_bij_betw_dom_img_if_ground_img: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "ground (subst_range \<sigma>)" shows "inj \<sigma> = bij_betw \<sigma> (subst_domain \<sigma>) (subst_range \<sigma>)"
Quotient_rel_abs2: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and T :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'a" assumes "Quotient R Abs Rep T"   and "R (Rep x) y" shows "x = Abs y"
simps_1: fixes e :: "nat \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a list \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a list \<Rightarrow> bool" shows "eval e f g \<bottom> = False"
AbstrLevels_A3_A31: fixes i :: "AbstrLevelsID" assumes "sA3 \<in> AbstrLevel i" shows "sA31 \<notin> AbstrLevel i"
assms_3: shows "fst (rep_nat_term u) = fst (rep_nat_term a)"
fixp_induct: fixes P :: "('c \<Rightarrow> 'd llist) \<Rightarrow> bool"   and g :: "'c \<Rightarrow> ('d, 'c) step" assumes "llist.admissible P"   and "P (\<lambda>lunstream. LNil)"   and "\<And>s. \<lbrakk>P s; llist.le_fun s (lunstream g)\<rbrakk> \<Longrightarrow> P (\<lambda>sa. case g sa of Done \<Rightarrow> LNil | Skip s' \<Rightarrow> s s' | Yield x s' \<Rightarrow> LCons x (s s'))" shows "P (lunstream g)"
simps_1: fixes W :: "nat set" shows "unforwarded_non_volatile_reads [] W = {}"
subst_comp_num: fixes \<phi> :: "'fmla"   and x :: "'var"   and y :: "'var"   and n :: "'trm" assumes "\<phi> \<in> fmla"   and "x \<in> var"   and "y \<in> var"   and "n \<in> num"   and "x \<noteq> y" shows "subst (subst \<phi> (Var x) y) n x = subst (subst \<phi> n x) n y"
induct: fixes x1 :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and x2 :: "com"   and x3 :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and P :: "((char list \<Rightarrow> int) \<Rightarrow> bool) \<Rightarrow> com \<Rightarrow> ((char list \<Rightarrow> int) \<Rightarrow> bool) \<Rightarrow> bool" assumes "\<turnstile>\<^sub>t {x1} x2 {x3}"   and "\<And>P. P P SKIP P"   and "\<And>P a x. P (\<lambda>s. P (s[a/x])) (x ::= a) P"   and "\<And>P\<^sub>1 c\<^sub>1 P\<^sub>2 c\<^sub>2 P\<^sub>3. \<lbrakk>\<turnstile>\<^sub>t {P\<^sub>1} c\<^sub>1 {P\<^sub>2}; P P\<^sub>1 c\<^sub>1 P\<^sub>2; \<turnstile>\<^sub>t {P\<^sub>2} c\<^sub>2 {P\<^sub>3}; P P\<^sub>2 c\<^sub>2 P\<^sub>3\<rbrakk> \<Longrightarrow> P P\<^sub>1 (c\<^sub>1;; c\<^sub>2) P\<^sub>3"   and "\<And>P b c\<^sub>1 Q c\<^sub>2. \<lbrakk>\<turnstile>\<^sub>t {\<lambda>s. P s \<and> bval b s} c\<^sub>1 {Q}; P (\<lambda>s. P s \<and> bval b s) c\<^sub>1 Q; \<turnstile>\<^sub>t {\<lambda>s. P s \<and> \<not> bval b s} c\<^sub>2 {Q}; P (\<lambda>s. P s \<and> \<not> bval b s) c\<^sub>2 Q\<rbrakk> \<Longrightarrow> P P (IF b THEN c\<^sub>1 ELSE c\<^sub>2) Q"   and "\<And>P b T c. \<lbrakk>\<And>n. \<turnstile>\<^sub>t {\<lambda>s. P s \<and> bval b s \<and> T s n} c {\<lambda>s. P s \<and> (\<exists>n'<n. T s n')}; \<And>n. P (\<lambda>s. P s \<and> bval b s \<and> T s n) c (\<lambda>s. P s \<and> (\<exists>n'<n. T s n'))\<rbrakk> \<Longrightarrow> P (\<lambda>s. P s \<and> Ex (T s)) (WHILE b DO c) (\<lambda>s. P s \<and> \<not> bval b s)"   and "\<And>P' P c Q Q'. \<lbrakk>\<forall>s. P' s \<longrightarrow> P s; \<turnstile>\<^sub>t {P} c {Q}; P P c Q; \<forall>s. Q s \<longrightarrow> Q' s\<rbrakk> \<Longrightarrow> P P' c Q'" shows "P x1 x2 x3"
r_cancel: fixes prod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and y :: "'a"   and x :: "'a"   and z :: "'a" assumes "Dgrp prod one" shows "(prod y x = prod z x) = (y = z)"
A\<^sub>i\<^sub>i\<^sub>i: fixes x :: "i"   and y :: "i"   and z :: "i" shows "x \<cdot> (y \<cdot> z) \<cong> (x \<cdot> y) \<cdot> z"
span_in_category_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b span_data" shows "span_in_category C S \<equiv> category C \<and> span_in_category_axioms C S"
xAb: shows "is_solution x (fst (solve_system A b)) (snd (solve_system A b))"
dataflow_topology_def: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c" shows "dataflow_topology summary results_in \<equiv> graph summary \<and> dataflow_topology_axioms summary results_in"
final_subdivFace': fixes g :: "graph"   and f :: "face"   and r :: "nat"   and u :: "nat"   and n :: "nat"   and ovs :: "nat option list" assumes "minGraphProps g"   and "pre_subdivFace' g f r u n ovs"   and "f \<in> \<F> g"   and "ovs = [] \<longrightarrow> n = 0 \<and> u = last (verticesFrom f r)" shows "\<exists>f'\<in>set (finals (subdivFace' g f u n ovs)) - set (finals g). (f\<^bsup>-1\<^esup> \<bullet> r, r) \<in> \<E> f' \<and> |vertices f'| = n + |ovs| + (if r = u then 1 else |between (vertices f) r u| + 2)"
mset_nth_drop_less: fixes i :: "nat"   and xs :: "'a list" assumes "i < length xs" shows "mset (nth_drop i xs) < mset xs"
F2rel_cong: fixes R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and S1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and S2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R3 :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and S3 :: "'h \<Rightarrow> 'i \<Rightarrow> bool" assumes "R1 = S1"   and "R2 = S2"   and "R3 = S3" shows "F2rel R1 R2 R3 = F2rel S1 S2 S3"
consequence_graphD_3: fixes Rs :: "(('e, 'f) labeled_graph \<times> ('e, 'f) labeled_graph) set"   and G :: "('e, 'g) labeled_graph" assumes "consequence_graph Rs G" shows "graph G"
FL_equivalence_implies_bisimilarity: fixes L :: "'b \<times> 'c fs_set \<times> 'c \<Rightarrow> 'c fs_set"   and satisfies :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and transition :: "'d \<Rightarrow> ('b, 'd) residual \<Rightarrow> bool"   and effect_apply :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c fs_set"   and P :: "'d"   and Q :: "'d" assumes "indexed_effect_nominal_ts TYPE('a) L satisfies transition effect_apply"   and "indexed_effect_nominal_ts.FL_logically_equivalent TYPE('a) L satisfies transition effect_apply F P Q" shows "effect_nominal_ts.FL_bisimilar satisfies transition effect_apply L F P Q"
prems_2: shows "ord_rec_Seq T G s (succ k) y"
case_cong: fixes sum :: "'h + 'i"   and sum' :: "'h + 'i"   and f1 :: "'h \<Rightarrow> 'j"   and g1 :: "'h \<Rightarrow> 'j"   and f2 :: "'i \<Rightarrow> 'j"   and g2 :: "'i \<Rightarrow> 'j" assumes "sum = sum'"   and "\<And>x1. sum' = Inl x1 \<Longrightarrow> f1 x1 = g1 x1"   and "\<And>x2. sum' = Inr x2 \<Longrightarrow> f2 x2 = g2 x2" shows "(case sum of Inl x1 \<Rightarrow> f1 x1 | Inr x2 \<Rightarrow> f2 x2) = (case sum' of Inl x1 \<Rightarrow> g1 x1 | Inr x2 \<Rightarrow> g2 x2)"
that: fixes p :: "(name \<times> name) list"   and \<Psi>' :: "'b"   and \<Psi>\<^sub>P' :: "'b"   and A\<^sub>P' :: "name list" assumes "set p \<subseteq> set (bn \<alpha>) \<times> set (bn (p \<bullet> \<alpha>))"   and "(p \<bullet> \<Psi>\<^sub>P) \<otimes> \<Psi>' \<simeq> \<Psi>\<^sub>P'"   and "distinctPerm p"   and "extractFrame P' = \<langle>A\<^sub>P', \<Psi>\<^sub>P'\<rangle>"   and "A\<^sub>P' \<sharp>* P'"   and "A\<^sub>P' \<sharp>* \<alpha>"   and "A\<^sub>P' \<sharp>* (p \<bullet> \<alpha>)"   and "A\<^sub>P' \<sharp>* C"   and "bn (p \<bullet> \<alpha>) \<sharp>* C'"   and "bn (p \<bullet> \<alpha>) \<sharp>* \<alpha>"   and "bn (p \<bullet> \<alpha>) \<sharp>* P'"   and "distinct A\<^sub>P'" shows "thesis"
goodBinp_imp_qGoodBinp_pickBinp: fixes binp :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qAbs set option" assumes "goodBinp binp" shows "qGoodBinp (pickBinp binp)"
prefix_bottom_iff: fixes xs :: "[:'r:]"   and ys :: "[:'r:]" shows "(prefix\<cdot>xs\<cdot>ys = \<bottom>) = (xs = \<bottom> \<or> ys = \<bottom>)"
A2: shows "\<bar>spmf (R1_14_interm1 M (c0, c1) \<bind> D) True - spmf (R1_14_interm2 M (c0, c1) \<bind> D) True\<bar> = \<bar>spmf (pair_spmf coin_spmf coin_spmf \<bind> (\<lambda>(m0, m1). R1_OT12 (m0, m1) c1 \<bind> (\<lambda>view. A2 view (m0, m1)))) True - spmf (pair_spmf coin_spmf coin_spmf \<bind> (\<lambda>(m0, m1). S1_OT12 (m0, m1) () \<bind> (\<lambda>view. A2 view (m0, m1)))) True\<bar>"
subring_Ring: fixes S :: "('a, 'd) Ring_scheme" assumes "Subring R S" shows "Ring S"
simps_3: fixes x :: "(int \<times> int) set set"   and y :: "(int \<times> int) set set"   and p :: "int" shows "C (AllowPortFromTo x y p) = allow_from_to_port p x y"
wt: shows "P,E \<turnstile> e # es [::] Ts"
evenperm_comp: fixes p :: "'b \<Rightarrow> 'b"   and q :: "'b \<Rightarrow> 'b" assumes "permutation p"   and "permutation q" shows "evenperm (p \<circ> q) = (evenperm p = evenperm q)"
right_neutral: fixes a :: "'d upred" shows "(a \<and> true) = a"
rel_gpv''_conversep: fixes A :: "'j \<Rightarrow> 'g \<Rightarrow> bool"   and C :: "'k \<Rightarrow> 'h \<Rightarrow> bool"   and R :: "'l \<Rightarrow> 'i \<Rightarrow> bool" shows "rel_gpv'' A\<inverse>\<inverse> C\<inverse>\<inverse> R\<inverse>\<inverse> = (rel_gpv'' A C R)\<inverse>\<inverse>"
2: fixes v :: "'a vec"   and n :: "nat"   and a :: "'a" assumes "v \<in> carrier_vec n"   and "P n v" shows "P (Suc n) (vCons a v)"
lt_csucc_iff: fixes \<alpha> :: "V"   and \<kappa> :: "V" assumes "Ord \<alpha>"   and "Card \<kappa>" shows "(\<alpha> < csucc \<kappa>) = (vcard \<alpha> \<le> \<kappa>)"
isGlb_def: fixes R :: "'b set"   and S :: "'b set"   and x :: "'b" shows "isGlb R S x = greatestP (isLb R S) x"
induct: fixes x1 :: "'d rexp"   and x2 :: "'d rexp"   and P :: "'d rexp \<Rightarrow> 'd rexp \<Rightarrow> bool" assumes "x1 ~ x2"   and "\<And>r s t. P (Alt (Alt r s) t) (Alt r (Alt s t))"   and "\<And>r s t. P (Alt r (Alt s t)) (Alt (Alt r s) t)"   and "\<And>r s. P (Alt r s) (Alt s r)"   and "\<And>r. P r (Alt r r)"   and "\<And>r. P r r"   and "\<And>r r' s s'. \<lbrakk>r ~ r'; P r r'; s ~ s'; P s s'\<rbrakk> \<Longrightarrow> P (Alt r s) (Alt r' s')"   and "\<And>r r' s s'. \<lbrakk>r ~ r'; P r r'; s ~ s'; P s s'\<rbrakk> \<Longrightarrow> P (Conc r s) (Conc r' s')"   and "\<And>r r'. \<lbrakk>r ~ r'; P r r'\<rbrakk> \<Longrightarrow> P (Star r) (Star r')" shows "P x1 x2"
const_name_def: fixes t :: "'a" shows "const_name t = (case unconst t of Some name \<Rightarrow> name)"
sup_le1: fixes a :: "'a"   and b :: "'a" shows "a \<le> a \<squnion> b"
simps_2: fixes y :: "name"   and e :: "trm"   and \<theta> :: "(name \<times> trm) list"   and x :: "name" shows "(y, e) # \<theta> maps x to (if x = y then e else lookup \<theta> x)"
map_types_id: fixes t :: "Core.term" shows "map_types id t = t"
power_inverse: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and n :: "nat" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "power.power one times (inverse a) n = inverse (power.power one times a n)"
ih_22: shows "InvariantVarsQ (getQ state') F0 Vbl"
hyp_port_for'_def: fixes t :: "('form, 'rule, 'subst, 'var) itnode rose_tree"   and "is" :: "nat list"   and f :: "'form" shows "hyp_port_for' t is f = (SOME x. case x of (is', i, h) \<Rightarrow> prefix (is' @ [i]) is \<and> i < length (inPorts' (iNodeOf (tree_at t is'))) \<and> hyps (iNodeOf (tree_at t is')) h = Some (inPorts' (iNodeOf (tree_at t is')) ! i) \<and> f = subst (iSubst (tree_at t is')) (freshen (iAnnot (tree_at t is')) (labelsOut (iNodeOf (tree_at t is')) h)))"
JN_absorb: fixes k :: "'c"   and I :: "'c set"   and F :: "'c \<Rightarrow> 'd program" assumes "k \<in> I" shows "F k \<squnion> JOIN I F = JOIN I F"
rScope: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and x :: "name"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> \<tau> \<prec> P'"   and "\<And>C. Prop C \<Psi> P P'"   and "x \<sharp> \<Psi>"   and "x \<sharp> C" shows "Prop C \<Psi> (\<lparr>\<nu>x\<rparr>P) (\<lparr>\<nu>x\<rparr>P')"
EqRel: fixes \<Psi>' :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "Eq \<Psi>' P Q" shows "(\<Psi>', P, Q) \<in> Rel"
simple_state_rel_def: fixes erel :: "('f \<times> 'g) set" shows "\<langle>erel\<rangle>simple_state_rel \<equiv> {(s, s'). ss_stack s = map (\<lambda>u. (u, pending s' `` {u})) (stack s') \<and> on_stack s = set (stack s') \<and> visited s = dom (discovered s') \<and> dom (finished s') = dom (discovered s') - set (stack s') \<and> set (stack s') \<subseteq> dom (discovered s') \<and> (simple_state.more s, state.more s') \<in> erel}"
of_nat_le_iff: fixes one :: "'g"   and times :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and m :: "nat"   and n :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n) = (m \<le> n)"
correct0: fixes e :: "'e"   and x0 :: "'g"   and w0 :: "'h"   and x1 :: "'a" assumes "e \<in> carrier L"   and "(x0, w0) \<in> Rel0"   and "x1 \<in> valid_pub1" shows "\<Sigma>_OR.completeness_game (x0, x1) (Inl w0) e = return_spmf True"
excess_nz_iff_gz: fixes u :: "nat" assumes "u \<in> V"   and "u \<noteq> s" shows "(excess f u \<noteq> (0::'a)) = ((0::'a) < excess f u)"
lprefix_llength_le: fixes xs :: "'a llist"   and ys :: "'a llist" assumes "lprefix xs ys" shows "llength xs \<le> llength ys"
scale_right_imp_eq: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'d"   and a :: "'c"   and b :: "'c" assumes "vector_space scale"   and "x \<noteq> (0::'d)"   and "scale a x = scale b x" shows "a = b"
resample_fixp_induct: fixes P :: "('b set \<Rightarrow> 'b set \<Rightarrow> 'b spmf) \<Rightarrow> bool" assumes "spmf.admissible (\<lambda>resample. P (curry resample))"   and "P (\<lambda>resample A. return_pmf None)"   and "\<And>B. P B \<Longrightarrow> P (\<lambda>a b. spmf_of_set a \<bind> (\<lambda>x. if x \<in> b then return_spmf x else B a b))" shows "P resample"
univ_poly_is_cring: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set" assumes "domain R"   and "subring K R" shows "cring (K [X]\<^bsub>R\<^esub>)"
aliveImpl_def: fixes x :: "Value"   and s :: "StoreImpl" shows "aliveImpl x s = (case x of objV C a \<Rightarrow> a < newOID s C | arrV T a \<Rightarrow> a < newAID s T | _ \<Rightarrow> True)"
elims_2: fixes x :: "'d list"   and xa :: "'d list set"   and xb :: "'d list" assumes "mcp x xa xb" obtains z :: "'d list" and W :: "'d list set" and p :: "'d list" where "x = z" and "xa = W" and "xb = p" and "prefix p z \<and> p \<in> W \<and> (\<forall>p'. prefix p' z \<and> p' \<in> W \<longrightarrow> length p' \<le> length p)"
Ana_subst: fixes t :: "('fun, 'var) Term.term"   and K :: "('fun, 'var) Term.term list"   and R :: "('fun, 'var) Term.term list"   and \<delta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "Ana t = (K, R)"   and "K \<noteq> [] \<or> R \<noteq> []" shows "Ana (t \<cdot> \<delta>) = (K \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<delta>, R \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<delta>)"
rPar2: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>P :: "'b"   and Q :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and Q' :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and A\<^sub>Q :: "name list"   and \<Psi>\<^sub>Q :: "'b"   and C :: "'d" assumes "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> \<alpha> \<prec> Q'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<And>C. Prop C (\<Psi> \<otimes> \<Psi>\<^sub>P) Q (\<alpha> \<prec> Q') A\<^sub>Q \<Psi>\<^sub>Q"   and "distinct (bn \<alpha>)"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* \<alpha>"   and "A\<^sub>P \<sharp>* Q'"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* \<alpha>"   and "A\<^sub>Q \<sharp>* Q'"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* Q"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "bn \<alpha> \<sharp>* \<Psi>\<^sub>P"   and "bn \<alpha> \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>Q \<sharp>* C"   and "bn \<alpha> \<sharp>* C" shows "Prop C \<Psi> (P \<parallel> Q) (\<alpha> \<prec> P \<parallel> Q') (A\<^sub>P @ A\<^sub>Q) (\<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q)"
simps_1: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and at :: "('d, 'e) atm" assumes "ProblemIk wtFsym wtPsym arOf resOf parOf \<Phi> infTp" shows "ProblemIk.nvL (Pos at) = ProblemIk.nvA at"
rbt_greater_trans: fixes x :: "'a"   and y :: "'a"   and t :: "('a, 'c) rbt" assumes "x < y"   and "y \<guillemotleft>| t" shows "x \<guillemotleft>| t"
ide_trg: fixes \<mu> :: "'c" assumes "C'.arr \<mu>" shows "C'.ide (C'.trg \<mu>)"
disjoint_def: fixes A :: "'a set"   and B :: "'a set" shows "??.WordInterval.disjoint A B \<equiv> A \<inter> B = {}"
alt_from_ord_def: fixes L :: "'b list" shows "alt_from_ord L \<equiv> \<lambda>n. if n < length L then Some (L ! n) else None"
rep_eq: fixes prod :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and x :: "'d" assumes "bounded_bilinear prod" shows "blinfun_apply (bounded_bilinear.prod_right prod x) = prod x"
untilE: fixes t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'BB"   and n :: "nat"   and \<gamma>' :: "(nat \<Rightarrow> 'BB) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma> :: "(nat \<Rightarrow> 'BB) \<Rightarrow> nat \<Rightarrow> bool" assumes "pb.eval the_bb t t' n (\<gamma>' \<UU>\<^sub>b \<gamma>)" shows "\<exists>n'\<ge>n. pb.eval the_bb t t' n' \<gamma> \<and> (\<forall>n''\<ge>n. n'' < n' \<longrightarrow> pb.eval the_bb t t' n'' \<gamma>')"
space_pair_measure: fixes A :: "'c measure"   and B :: "'d measure" shows "space (A \<Otimes>\<^sub>M B) = space A \<times> space B"
simps_5: fixes f1 :: "'a"   and f2 :: "'a" shows "old.rec_bool f1 f2 True = f1"
natLeq_def: shows "natLeq \<equiv> {(x, y). x \<le> y}"
csl: shows "cs\<^bsup>\<pi>\<^esup> l = cs\<^bsup>\<pi>\<^esup> k @ ys @ [\<pi> l]"
aw_c: shows "snd (mus_adjuster f n_gs [] (0\<^sub>v n)) \<in> carrier_vec n"
rwof_rel_def: fixes init :: "'c nres"   and cond :: "'c \<Rightarrow> bool"   and step :: "'c \<Rightarrow> 'c nres" shows "rwof_rel init cond step \<equiv> {(s, s'). rwof init cond step s \<and> cond s \<and> inres (step s) s'}"
vE: fixes L :: "(nat \<times> int \<times> nat) list" shows "valid_graph (GD_from_list_\<alpha>_graph L L)"
pow_mult_distrib: fixes x :: "'a"   and y :: "'a"   and n :: "nat" assumes "x \<otimes> y = y \<otimes> x"   and "x \<in> carrier \<G>"   and "y \<in> carrier \<G>" shows "(x \<otimes> y) [^] n = x [^] n \<otimes> y [^] n"
trace_outer_prod_right: fixes A :: "'c mat"   and n :: "nat"   and v :: "'c vec"   and w :: "'c vec" assumes "A \<in> carrier_mat n n"   and "v \<in> carrier_vec n"   and "w \<in> carrier_vec n" shows "trace (A * outer_prod v w) = inner_prod w (A *\<^sub>v v)"
csqrt_minus: fixes x :: "complex" assumes "Im x < 0 \<or> Im x = 0 \<and> 0 \<le> Re x" shows "csqrt (- x) = \<i> * csqrt x"
convex_real_interval_1: fixes a :: "real" shows "convex {a..}"
hom_inv: shows "aG.digraph_isomorphism (inv_iso hom)"
HomFtorMap_def: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d"   and g :: "'e" shows "Hom\<^bsub>C\<^esub>[X,g] \<equiv> ZFfun (Hom\<^bsub>C\<^esub> X dom\<^bsub>C\<^esub> g) (Hom\<^bsub>C\<^esub> X cod\<^bsub>C\<^esub> g) (\<lambda>f. m2z\<^bsub>C\<^esub>z2m\<^bsub>C\<^esub>f ;;\<^bsub>C\<^esub> g)"
weak_ring_morphism_range: fixes f :: "('c multiset \<Rightarrow> 'a) list \<Rightarrow> 'd"   and I :: "('c multiset \<Rightarrow> 'a) list set"   and a :: "('c multiset \<Rightarrow> 'a) list" assumes "weak_ring_morphism f I (poly_ring L)"   and "a \<in> carrier (poly_ring L)" shows "f ` (I +>\<^bsub>poly_ring L\<^esub> a) = {f a}"
prod_mOp_def: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme" shows "prod_mOp I A = (\<lambda>f\<in>carr_prodag I A. \<lambda>x\<in>I. -\<^sub>a\<^bsub>A x\<^esub> f x)"
iu_condition_imply_secure: fixes P :: "'a process"   and I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b" assumes "ref_union_closed P"   and "weakly_future_consistent P I D (rel_ipurge P I D)" shows "secure P I D"
T_on_rand'_append: fixes A :: "('c list \<Rightarrow> 'd pmf) \<times> ('c list \<times> 'd \<Rightarrow> 'c \<Rightarrow> ((nat \<times> nat list) \<times> 'd) pmf)"   and s :: "('c list \<times> 'd) pmf"   and xs :: "'c list"   and ys :: "'c list" shows "Move_to_Front.T_on_rand' A s (xs @ ys) = Move_to_Front.T_on_rand' A s xs + Move_to_Front.T_on_rand' A (Partial_Cost_Model.config'_rand A s xs) ys"
hyps_2: fixes l :: "nat"   and q :: "'a"   and k :: "nat"   and p :: "'a" assumes "u = (l, q)"   and "v = (k, p)" shows "q \<in> reachable A p"
one_side_symmetry: fixes P :: "'p"   and Q :: "'p"   and A :: "'p"   and B :: "'p" assumes "P Q OS A B" shows "P Q OS B A"
pconj_less_one: fixes a :: "real"   and b :: "real" assumes "a + b < 1" shows "a .& b = 0"