hcju/isabelleps huggingface dataset:
subsetClosed: fixes p :: "(name \<times> name) list"   and S :: "'d set"   and T :: "'d set" shows "(p \<bullet> S \<subseteq> p \<bullet> T) = (S \<subseteq> T)"
content_times_primitive_part: fixes p :: "'a poly" shows "smult (content p) (primitive_part p) = p"
cong_sym: fixes b :: "'a"   and c :: "'a"   and a :: "'a" assumes "[b = c] (mod a)" shows "[c = b] (mod a)"
assms_4: shows "subst_domain \<I> = \<V> \<and> ground (subst_range \<I>)"
comm_neutral: fixes f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and z :: "'d"   and a :: "'d" assumes "comm_monoid f z" shows "f a z = a"
ex_of_int_less: fixes x :: "'a" shows "\<exists>z. of_int z < x"
swap: fixes P :: "bool"   and R :: "bool" assumes "\<not> P"   and "\<not> R \<Longrightarrow> P" shows "R"
assoc_iff_assoc_mult: fixes a :: "'a list"   and b :: "'a list" assumes "a \<in> carrier (K [X])"   and "b \<in> carrier (K [X])" shows "a \<sim>\<^bsub>K [X]\<^esub> b = a \<sim>\<^bsub>Ring_Divisibility.mult_of (K [X])\<^esub> b"
get_roots_UN_get_work_locs_subseteq_ghost_honorary_grey_empty_locs: shows "get_roots_UN_get_work_locs \<subseteq> ghost_honorary_grey_empty_locs"
sup1: fixes A :: "'a"   and k :: "'a"   and a :: "'a"   and b :: "'a" assumes "A \<equiv> k \<squnion> a" shows "A \<squnion> b \<equiv> k \<squnion> (a \<squnion> b)"
Oops2_not_AKcryptSK: fixes evs :: "event list"   and A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg"   and SK :: "nat" assumes "evs \<in> kerbIV"   and "Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace>) \<in> set evs" shows "\<not> AKcryptSK servK SK evs"
E29: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "|~ \<circle>\<diamond>F \<longrightarrow> \<diamond>F"
case_swap: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'a"   and p :: "'c \<times> 'b" shows "(case prod.swap p of (y, x) \<Rightarrow> f x y) = (case p of (x, y) \<Rightarrow> f x y)"
cases: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and a :: "'d" assumes "Imagep R P a" obtains x :: "'c" and y :: "'d" where "a = y" and "P x" and "R x y"
rComm2: fixes M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and K :: "'a"   and P'' :: "('a, 'b, 'c) psi"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "\<Psi> \<rhd> !P \<longmapsto> K\<lparr>N\<rparr> \<prec> P''"   and "\<Psi> \<turnstile> M \<leftrightarrow> K"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* K"   and "xvec \<sharp>* C" shows "Prop C \<Psi> (P \<parallel> !P) (\<lparr>\<nu>*xvec\<rparr>P' \<parallel> P'')"
\<phi>: shows "\<guillemotleft>\<phi> : g' \<Rightarrow>\<^sub>D F g\<guillemotright> \<and> D.iso \<phi>"
sel: fixes x :: "'b \<Rightarrow> 'c \<times> 'b" shows "run_state (State x) = x"
t_loop_5: shows "\<C>' \<turnstile> es : tfn _> tfm"
mset_del_min: fixes t :: "('b \<times> nat) tree" shows "mset_tree (del_min t) = mset_tree t - {#get_min t#}"
assms_2: shows "Lower_Semicontinuous.convex_on s f"
C\<^sub>i: shows "\<^bold>\<forall>x. \<^bold>\<not> (\<^bold>\<forall>xa. \<^bold>\<not> (((\<^bold>\<forall>x. xa \<cdot> x \<cong> x \<^bold>\<leftarrow> E (xa \<cdot> x)) \<^bold>\<and> (\<^bold>\<forall>x. x \<cdot> xa \<cong> x \<^bold>\<leftarrow> E (x \<cdot> xa))) \<^bold>\<and> xa \<cdot> x \<cong> x))"
word_of_nat_less: fixes n :: "nat"   and x :: "'b word" assumes "n < unat x" shows "word_of_nat n < x"
flatten_sum: fixes f :: "'d \<Rightarrow> 'b \<Rightarrow>\<^sub>0 'b \<Rightarrow>\<^sub>0 'c"   and I :: "'d set" shows "flatten (sum f I) = (\<Sum>i\<in>I. flatten (f i))"
similar_matrices_def: fixes A :: "(('a, 'b) vec, 'b) vec"   and B :: "(('a, 'b) vec, 'b) vec" shows "similar_matrices A B = (\<exists>P. invertible P \<and> B = matrix_inv P ** A ** P)"
interp_msg_def: fixes msg :: "'a \<times> operation"   and state :: "int" shows "interp_msg msg state \<equiv> counter_op (snd msg) state"
simps_2: fixes x :: "'b"   and hsl :: "'b hp list"   and hsr :: "'b hp list" shows "\<Phi>_hps (Hp x hsl # hsr) = \<Phi>_hps hsl + \<Phi>_hps hsr + log 2 (real (size_hps hsl + size_hps hsr + 1))"
keys_combine_with_key: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and m1 :: "('b, 'c) mapping"   and m2 :: "('b, 'c) mapping" shows "keys (combine_with_key f m1 m2) = keys m1 \<union> keys m2"
free_vars_marg_dens_cexpr: fixes vs :: "nat list"   and vs' :: "nat list"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "cexpr"   and x :: "nat" assumes "cdens_ctxt_invar vs vs' \<Gamma> \<delta>" shows "free_vars (marg_dens_cexpr \<Gamma> vs x \<delta>) \<subseteq> shift_var_set (set vs')"
elims_1: fixes x :: "('b, 'c, 'd) Tree"   and xa :: "('b, 'c, 'd) Tree"   and y :: "bool" assumes "(x =\<^sub>\<alpha> xa) = y" obtains tset1 :: "('b, 'c, 'd) Tree set['b]" and tset2 :: "('b, 'c, 'd) Tree set['b]" where "x = tConj tset1" and "xa = tConj tset2" and "y = rel_bset (=\<^sub>\<alpha>) tset1 tset2"   | t1 :: "('b, 'c, 'd) Tree" and t2 :: "('b, 'c, 'd) Tree" where "x = tNot t1__" and "xa = tNot t2__" and "y = (t1__ =\<^sub>\<alpha> t2__)"   | \<phi>1 :: "'c" and \<phi>2 :: "'c" where "x = tPred \<phi>1" and "xa = tPred \<phi>2" and "y = (\<phi>1 = \<phi>2)"   | \<alpha>1 :: "'d" and t1 :: "('b, 'c, 'd) Tree" and \<alpha>2 :: "'d" and t2 :: "('b, 'c, 'd) Tree" where "x = tAct \<alpha>1 t1" and "xa = tAct \<alpha>2 t2" and "y = (\<exists>p. (bn \<alpha>1, t1) \<approx>set (=\<^sub>\<alpha>) supp_rel (=\<^sub>\<alpha>) p (bn \<alpha>2, t2) \<and> (bn \<alpha>1, \<alpha>1) \<approx>set (=) supp p (bn \<alpha>2, \<alpha>2))"   | v :: "('b, 'c, 'd) Tree" and va :: "('b, 'c, 'd) Tree set['b]" where "x = tNot v__" and "xa = tConj va__" and "\<not> y"   | v :: "('b, 'c, 'd) Tree" and va :: "'c" where "x = tNot vb__" and "xa = tPred vaa__" and "\<not> y"   | v :: "('b, 'c, 'd) Tree" and va :: "'d" and vb :: "('b, 'c, 'd) Tree" where "x = tNot vc__" and "xa = tAct vab__ vba__" and "\<not> y"   | v :: "'c" and va :: "('b, 'c, 'd) Tree set['b]" where "x = tPred vd__" and "xa = tConj vac__" and "\<not> y"   | v :: "'c" and va :: "('b, 'c, 'd) Tree" where "x = tPred ve__" and "xa = tNot vad__" and "\<not> y"   | v :: "'c" and va :: "'d" and vb :: "('b, 'c, 'd) Tree" where "x = tPred vf__" and "xa = tAct vae__ vbb__" and "\<not> y"   | v :: "'d" and va :: "('b, 'c, 'd) Tree" and vb :: "('b, 'c, 'd) Tree set['b]" where "x = tAct vg__ vaf__" and "xa = tConj vbc__" and "\<not> y"   | v :: "'d" and va :: "('b, 'c, 'd) Tree" and vb :: "('b, 'c, 'd) Tree" where "x = tAct vh__ vag__" and "xa = tNot vbd__" and "\<not> y"   | v :: "'d" and va :: "('b, 'c, 'd) Tree" and vb :: "'c" where "x = tAct vi__ vah__" and "xa = tPred vbe__" and "\<not> y"   | va :: "('b, 'c, 'd) Tree set['b]" and v :: "('b, 'c, 'd) Tree" where "x = tConj vai__" and "xa = tNot vj__" and "\<not> y"   | va :: "('b, 'c, 'd) Tree set['b]" and v :: "'c" where "x = tConj vaj__" and "xa = tPred vk__" and "\<not> y"   | vb :: "('b, 'c, 'd) Tree set['b]" and v :: "'d" and va :: "('b, 'c, 'd) Tree" where "x = tConj vb" and "xa = tAct v va" and "\<not> y"
hyperpow_realpow: fixes r :: "real"   and n :: "nat" shows "hypreal_of_real r pow hypnat_of_nat n = hypreal_of_real (r ^ n)"
sumlist_carrier: fixes xs :: "'a list" assumes "set xs \<subseteq> carrier G" shows "sumlist xs \<in> carrier G"
g_3: fixes g :: "int poly" assumes "g \<in> set gs" shows "degree_m g = degree g"
progress_set_Union: fixes F :: "'b program"   and A :: "'b set"   and B' :: "'b set"   and C :: "'b set set"   and T :: "'b set"   and B :: "'b set"   and G :: "'b program" assumes "F \<in> A \<longmapsto> B'"   and "C \<in> progress_set F T B"   and "F \<in> stable T"   and "B \<subseteq> B'"   and "B' \<in> C"   and "\<And>X. X \<in> C \<Longrightarrow> G \<in> X - B co X" shows "F \<squnion> G \<in> T \<inter> A \<longmapsto> B'"
\<sigma>_ineqs_fv_dom': fixes X :: "'var list"   and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and \<delta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t \<in> set S"   and "subst_domain \<delta> = set X \<and> ground (subst_range \<delta>)" shows "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<delta>) \<subseteq> subst_domain \<sigma>"
alphaAbs_all_qFresh_def: fixes xs :: "'d"   and x :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm"   and xs' :: "'d"   and x' :: "'e"   and X' :: "('f, 'g, 'd, 'e, 'h) qTerm" shows "alphaAbs_all_qFresh xs x X xs' x' X' \<equiv> xs = xs' \<and> (\<forall>y. qFresh xs y X \<and> qFresh xs y X' \<longrightarrow> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
sup_fun_def: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "sup f g = (\<lambda>x. sup (f x) (g x))"
create_graph_impl_def: shows "create_graph_impl \<equiv> \<lambda>x. expand_impl (\<lparr>name_impl = Suc 0, incoming_impl = map2set_insert rbt_insert 0 rbt.Empty, new_impl = lss.ins_dj x (lss.empty ()), old_impl = lss.empty (), next_impl = lss.empty ()\<rparr>, []) \<bind> (\<lambda>(xa, y). RETURN y)"
natural_transformation_axioms: shows "natural_transformation A B G K \<sigma>\<tau>.map"
arctan_boundaries: fixes x :: "float"   and prec :: "nat" shows "arctan (real_of_float x) \<in> {real_of_float (lb_arctan prec x)..real_of_float (ub_arctan prec x)}"
sel_4: fixes x1 :: "'c"   and x2 :: "'d"   and x3 :: "nat"   and x4 :: "('c, 'd) BinomialTree list" shows "children (Node x1 x2 x3 x4) = x4"
unord_resolve_counterex_reducing: fixes N :: "'c literal multiset set"   and C :: "'c literal multiset" assumes "{#} \<notin> N"   and "C \<in> N"   and "\<not> INTERP N \<Turnstile> C"   and "\<And>D. \<lbrakk>D \<in> N; \<not> INTERP N \<Turnstile> D\<rbrakk> \<Longrightarrow> C \<le> D" obtains D :: "'c literal multiset" and E :: "'c literal multiset" where "D \<in> N" and "INTERP N \<Turnstile> D" and "production N D \<noteq> {}" and "unord_resolve D C E" and "\<not> INTERP N \<Turnstile> E" and "E < C"
uc_par_meet: fixes x :: "'a"   and y :: "'a" shows "x \<parallel> U \<sqinter> y \<parallel> U = x \<parallel> U \<parallel> y \<parallel> U"
IfElim1: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and os :: "Val list"   and S :: "(Var \<times> Val) list"   and h :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list"   and n :: "nat"   and ll :: "int"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and pc :: "int" assumes "(M, l, (os, S, h), n, ll, t) \<in> Step"   and "get_ins M l = Some (iftrue pc)" shows "(\<exists>ops. os = TRUE # ops \<and> n = Suc 0 \<and> t = (ops, S, h) \<and> ll = pc) \<or> (\<exists>v ops. v \<noteq> TRUE \<and> os = v # ops \<and> n = Suc 0 \<and> t = (ops, S, h) \<and> ll = l + 1)"
input2state_out_of_bounds: fixes ia :: "value list"   and i :: "nat" assumes "length ia \<le> i" shows "input2state ia $ i = None"
infinite_arbitrarily_large: fixes A :: "'g set"   and n :: "nat" assumes "infinite A" shows "\<exists>B. finite B \<and> card B = n \<and> B \<subseteq> A"
alignUp_le_greater_al: fixes a :: "'b word"   and x :: "'b word"   and n :: "nat" assumes "a \<le> x"   and "n < LENGTH('b)"   and "is_aligned x n" shows "alignUp a n \<le> x"
add_decreasing2: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c \<le> (0::'a)"   and "a \<le> b" shows "a + c \<le> b"
rep_eq: fixes x :: "('b, 'c) pp" shows "lookup_pp x = lookup (pp.mapping_of x)"
KBasic_7: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<^bold>\<box>\<phi> \<^bold>& \<^bold>\<box>\<psi> \<^bold>\<rightarrow> \<^bold>\<box>(\<phi> \<^bold>\<equiv> \<psi>) in v]"
delta_hedging_def: fixes N :: "bool stream measure"   and der :: "bool stream \<Rightarrow> real"   and matur :: "nat" shows "delta_hedging N der matur = self_fin_delta_pf N der matur (integral\<^sup>L N (discounted_value r (\<lambda>m. der) matur))"
bool_to_ternary_pullup1: fixes X :: "bool" shows "eval_ternary_Not (bool_to_ternary X) = bool_to_ternary (\<not> X)"
l2_trans_def: shows "l2_trans \<equiv> \<Union>m M X Rb Ra A B K Y. (l2_step1 Ra A B \<union> l2_step2 Rb A B X Y \<union> l2_step3 Ra A B X Y \<union> l2_step4 Rb A B X Y \<union> l2_dy_fake_chan M \<union> l2_dy_fake_msg m \<union> l2_lkr_others A \<union> l2_lkr_after A \<union> l2_skr Ra K)\<^sup>="
red_OA_subset_black_OA: fixes prb :: "('d, 'e, 'f) pre_RedBlack"   and rv :: "'d \<times> nat" assumes "RedBlack prb" shows "ui_edge ` out_edges (red prb) rv \<subseteq> out_edges (black prb) (fst rv)"
distinct_12: fixes x41 :: "dbtm"   and x42 :: "dbtm"   and x3 :: "nat" shows "DBEats x41 x42 \<noteq> DBInd x3"
red_rtranclp_diff_in_pmdl: fixes F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" assumes "(punit.red F)\<^sup>*\<^sup>* p q" shows "p - q \<in> local.punit.pmdl F"
entails_iff_unsatisfiable_single: fixes CC :: "'b literal multiset set"   and E :: "'b literal multiset" shows "(CC \<TTurnstile>e {E}) = (\<not> satisfiable (CC \<union> {{#- L#} |L. L \<in># E}))"
cod_char: fixes f :: "'c \<times> 'c" shows "CopxC.cod f = (if Cop.arr (fst f) \<and> C.arr (snd f) then (Cop.cod (fst f), C.cod (snd f)) else (Cop.null, C.null))"
distinct_1: fixes x11 :: "trm"   and x12 :: "trm"   and x21 :: "fmla"   and x22 :: "fmla" shows "x11 EQ x12 \<noteq> x21 OR x22"
run\<^sub>t_foldl: fixes \<Delta> :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and q\<^sub>0 :: "'a"   and w :: "nat \<Rightarrow> 'b"   and i :: "nat" shows "run\<^sub>t \<Delta> q\<^sub>0 w i = (foldl \<Delta> q\<^sub>0 (map w [0..<i]), w i, foldl \<Delta> q\<^sub>0 (map w [0..<Suc i]))"
open_PiE: fixes X :: "'c \<Rightarrow> 'd set" assumes "\<And>i. open (X i)"   and "finite {i. X i \<noteq> UNIV}" shows "open (Pi\<^sub>E UNIV X)"
Esigma_properdiv_set: fixes m :: "nat" assumes "1 \<le> m" shows "Esigma m = aliquot_sum m + m"
full_pmdlD: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and K :: "nat set"   and v :: "'t" assumes "p \<in> full_pmdl K"   and "v \<in> keys p" shows "component_of_term v \<in> K"
subcocycle_AE_in_context: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and u :: "nat \<Rightarrow> 'b \<Rightarrow> real" assumes "mpt M T"   and "\<And>m n. AE x in M. u (n + m) x \<le> u n x + u m ((T ^^ n) x)"   and "\<And>n. integrable M (u n)" shows "\<exists>w. mpt.subcocycle M T w \<and> (AE x in M. \<forall>n. w n x = u n x)"
dominatesI: fixes m :: "'node"   and g :: "'g"   and n :: "'node" assumes "m \<in> set (\<alpha>n g)"   and "\<And>ns. g \<turnstile> Entry g-ns\<rightarrow>m \<Longrightarrow> n \<in> set ns" shows "dominates g n m"
det_interchange_rows: fixes A :: "(('c, 'd) vec, 'd) vec"   and i :: "'d"   and j :: "'d" shows "det (interchange_rows A i j) = of_int (if i = j then 1 else - 1) * det A"
angle_def: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "angle a b c = vangle (a - b) (c - b)"
sum_upto_dirichlet_prod: shows "sum_upto (dirichlet_prod (\<lambda>_. 1) f) x = sum_upto (\<lambda>d. 1 * sum_upto f (x / real d)) x"
prod_in_span: fixes b :: "'b vec"   and n :: "nat"   and S :: "'b vec set"   and a :: "'b" assumes "b \<in> carrier_vec n"   and "S \<subseteq> carrier_vec n"   and "a = (0::'b) \<or> b \<in> LinearCombinations.module.span class_ring (module_vec TYPE('b) n) S" shows "a \<cdot>\<^sub>v b \<in> LinearCombinations.module.span class_ring (module_vec TYPE('b) n) S"
all_zero_below_greatest: shows "\<forall>m\<ge>(GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'rows). A $ m $ mod_type_class.from_nat k = (0::'a)"
assms_1: shows "\<langle>b, a\<rangle> \<in>\<^sub>\<circ> r\<inverse>\<^sub>\<circ>"
nat_pow_Suc: fixes x :: "'a"   and n :: "nat" shows "x ^ Suc n = x ^ n * x"
sc: fixes r :: "sequent list \<times> sequent"   and R :: "(sequent list \<times> sequent) set"   and S :: "sequent" assumes "r \<in> R"   and "case r of (x, xa) \<Rightarrow> (x, xa) \<in> upRules" shows "extendRule S r \<in>  R*"
pair_in_Id_conv: fixes a :: "'d"   and b :: "'d" shows "((a, b) \<in> Id) = (a = b)"
elims_3: fixes x :: "('b, 'c) Combinators list" assumes "\<not> wellformed_policy1_strong x" obtains "x = []"   | xa :: "('b, 'c) Combinators" and xs :: "('b, 'c) Combinators list" where "x = xa # xs" and "\<not> (xa = DenyAll \<and> DenyAll \<notin> set xs)"
SUP_upper: fixes Inf :: "'k set \<Rightarrow> 'k"   and Sup :: "'k set \<Rightarrow> 'k"   and inf :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and sup :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and bot :: "'k"   and top :: "'k"   and i :: "'l"   and A :: "'l set"   and f :: "'l \<Rightarrow> 'k" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "i \<in> A" shows "less_eq (f i) (Sup (f ` A))"
rule_LR: fixes u :: "'a"   and D :: "'b \<Rightarrow> 'a"   and x :: "'b"   and I :: "('a \<times> 'a) set"   and P :: "'b process"   and xs :: "'b list" assumes "u \<in> range D"   and "(D x, u) \<notin> I"   and "x \<in> next_events P xs" shows "(u, xs, xs @ [x]) \<in> rel_induct_aux P I D"
assms: fixes x :: "'a" assumes "x \<in># A" shows "f x \<subseteq># g x"
elims_2: fixes x :: "'d"   and xa :: "'e"   and xb :: "('f, 'g, 'd, 'e, 'h) qAbs" assumes "qAFreshAbs x xa xb" obtains xs :: "'d" and x :: "'e" and ys :: "'d" and y :: "'e" and X :: "('f, 'g, 'd, 'e, 'h) qTerm" where "xa__ = xs" and "xa = x" and "xb = qAbs ys y X" and "(xs \<noteq> ys \<or> x \<noteq> y) \<and> qAFresh xs x X"
iprev_iMin: fixes I :: "nat set" shows "iprev (iMin I) I = iMin I"
cnj_sum: fixes f :: "'a \<Rightarrow> complex"   and s :: "'a set" shows "Complex.cnj (sum f s) = (\<Sum>x\<in>s. Complex.cnj (f x))"
wlsAbs_inversion2: fixes xs :: "'varSort"   and s :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set" assumes "wlsAbs (xs, s) A" shows "isInBar (xs, s) \<and> (\<exists>x X. wls s X \<and> A = Abs xs x X)"
distinct_11: fixes x3 :: "val option" shows "Abs x3 \<noteq> Wrong"
emeasure_T': fixes X :: "('a \<times> 'b) stream set"   and I :: "('a \<times> 'b) pmf" assumes "X \<in> sets (stream_space (count_space UNIV))" shows "emeasure (T' I) X = \<integral>\<^sup>+ s. emeasure (T s) {\<omega> \<in> space (stream_space (count_space UNIV)). s ## \<omega> \<in> X} \<partial>measure_pmf I"
bounds_1: fixes n :: "nat" shows "real_of_float (lb_arctan_horner prec n (2 * 0 + 1) y) \<le> (\<Sum>j = 0..<n. (- 1) ^ j * (1 / real (2 * (0 + j) + 1)) * real_of_float (y ^ j))"
rep_Gromov_completion_limit: fixes x :: "'b Gromov_completion" shows "(\<lambda>n. to_Gromov_completion (rep_Gromov_completion x n)) \<longlonglongrightarrow> x"
good_ctx_def: fixes \<Gamma> :: "(ty \<times> nat) list \<times> nat" shows "good_ctx \<Gamma> \<equiv> \<forall>n<length (fst \<Gamma>). snd (fst \<Gamma> ! n) \<le> snd \<Gamma>"
smult_assoc1: fixes a :: "'a"   and b :: "'a"   and x :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "x \<in> carrier P" shows "a \<otimes> b \<odot>\<^bsub>P\<^esub> x = a \<odot>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> x)"
hmem_hempty: fixes a :: "hf" shows "a \<^bold>\<notin> 0"
wf_cdecl_supD: fixes wf_md :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'b \<Rightarrow> bool"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and D :: "char list"   and r :: "(char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list" assumes "wf_cdecl wf_md P (C, D, r)"   and "C \<noteq> Object" shows "is_class P D"
vals_merge: fixes t :: "('d, 'c) bintree"   and r :: "('d, 'c) bintree" shows "set (bt_dfs val (merge t r)) = set (bt_dfs val t) \<union> set (bt_dfs val r)"
invariant_simplex'_def: shows "invariant_simplex' = Incremental_NS_Constraint_Ops_To_Ns_For_Incremental_Simplex.invariant_cs invariant_nsc to_ns"
simps: fixes var :: "nat"   and a :: "real mpoly"   and b :: "real mpoly"   and c :: "real mpoly"   and d :: "real mpoly"   and F :: "atom fm" shows "substInfinitesimalQuadratic_fm var a b c d F = liftmap (\<lambda>x. substInfinitesimalQuadratic (var + x) (liftPoly 0 x a) (liftPoly 0 x b) (liftPoly 0 x c) (liftPoly 0 x d)) F 0"
hfg: fixes i :: "int" shows "h i = f i + coeff_int (B * G) i"
iarray_update_def: fixes a :: "'a iarray"   and i :: "nat"   and x :: "'a" shows "iarray_update a i x = IArray.of_fun (\<lambda>j. if j = i then x else a !! j) (IArray.length a)"
simps_1: fixes i :: "'a" shows "extended_to_set \<lbrakk>i\<rbrakk> = {i}"
CardinalityArityTransformation_axioms: shows "CardinalityArityTransformation Aexp Aheap"
DirProds_one_iso: fixes G :: "'d"   and f :: "'d \<Rightarrow> ('e, 'f) monoid_scheme" shows "(\<lambda>x. x G) \<in> Group.iso (DirProds f {G}) (f G)"
intrinsic_border_take_less: fixes j :: "nat"   and w :: "'a list" assumes "0 < j"   and "w \<noteq> []" shows "length (intrinsic_border (take j w)) < length w"
elims: fixes x :: "('j, 'h, 'i) generat"   and y :: "('h \<times> 'i) option" assumes "dest_IO x = y" obtains uu_ :: "'j" where "x = Pure uu_" and "y = None"   | out :: "'h" and c :: "'i" where "x = IO out c" and "y = Some (out, c)"
agree_restrict_exec_idempot: fixes as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list"   and PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set"   and s' :: "('c, 'd) fmap"   and s :: "('c, 'd) fmap"   and vs :: "'c set" assumes "as \<in> valid_plans PROB"   and "s' \<in> valid_states PROB"   and "s \<in> valid_states PROB"   and "\<forall>a. ListMem a as \<longrightarrow> agree (fmrestrict_set vs (snd a)) (fmrestrict_set vs s)"   and "fmrestrict_set vs s' = fmrestrict_set vs s" shows "fmrestrict_set vs (exec_plan s' as) = fmrestrict_set vs s"
to_st_trms_SMP_subset: fixes A :: "('fun, 'var) extstrand_step list" shows "trms\<^sub>s\<^sub>t (to_st A) \<subseteq> SMP (\<Union> (trms\<^sub>e\<^sub>s\<^sub>t\<^sub>p ` set A))"
intro: fixes hom :: "'c \<Rightarrow> 'd" assumes "\<And>x y. hom (x * y) = hom x * hom y" shows "times_hom hom"
of_nat_div: fixes m :: "nat"   and n :: "nat" shows "of_nat (m div n) = of_nat m div of_nat n"
leq: shows "(\<Sum>(r, c)\<leftarrow>C. lec_of_constraint (r *R c)) = Le_Constraint rel 0 d"
Eq\<^sub>1_prop: fixes v :: "i"   and F :: "\<Pi>\<^sub>1"   and G :: "\<Pi>\<^sub>1"   and \<phi> :: "\<Pi>\<^sub>1 \<Rightarrow> \<o>" assumes "[F \<^bold>=\<^sub>1 G in v]" shows "[\<phi> F in v] = [\<phi> G in v]"
assms_2: fixes x :: "'a" assumes "eval B x" shows "eval A x"
inject: fixes x1 :: "('d \<times> 'e \<times> 'e) set"   and x2 :: "'e set"   and y1 :: "('d \<times> 'e \<times> 'e) set"   and y2 :: "'e set" shows "(LG x1 x2 = LG y1 y2) = (x1 = y1 \<and> x2 = y2)"
rPar2: fixes \<Psi>\<^sub>P :: "'b"   and Q' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and A\<^sub>Q :: "name list"   and \<Psi>\<^sub>Q :: "'b" assumes "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>P"   and "distinct A\<^sub>Q"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* M"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P\<^sub>Q = A\<^sub>P @ A\<^sub>Q"   and "\<Psi>\<^sub>P\<^sub>Q = \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q" shows "Prop (P \<parallel> Q')"
bit_rec: fixes a :: "'a"   and n :: "nat" shows "bit a n = (if n = 0 then odd a else bit (a div (2::'a)) (n - 1))"