hcju/isabelleps huggingface dataset:
assms_5: shows "\<forall>\<^sub>F x in F. g x \<in> {l2 x..u2 x}"
assms: shows "\<lparr>s;vs;[$Call j]\<rparr> \<leadsto>_ i \<lparr>s';vs';es'\<rparr>"
generator: fixes x :: "('b, 'a) generator" shows "generator x \<in> {g. terminates g}"
Cinf_def: shows "C\<^sup>\<infinity> = fix\<cdot>C"
borel_integrable_compact: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "compact S"   and "continuous_on S f" shows "integrable lborel (\<lambda>x. indicat_real S x *\<^sub>R f x)"
distinct_mtf2: fixes n :: "nat"   and x :: "'c"   and xs :: "'c list" shows "distinct (mtf2 n x xs) = distinct xs"
simps_2: fixes h :: "'a"   and t :: "'a list"   and P :: "'a \<Rightarrow> bool" shows "lookup (h # t) P = (if P h then Some h else lookup t P)"
distinct_15: fixes x31 :: "nat"   and x32 :: "'a"   and x33 :: "'a"   and x34 :: "'b"   and x35 :: "'b"   and x36 :: "'c"   and x4 :: "'a" shows "Recv x31 x32 x33 x34 x35 x36 \<noteq> Snapshot x4"
preserves_hom: fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "Cop.in_hom f a b" shows "\<guillemotleft>Cones.map f : Cones.map a \<rightarrow>\<^sub>S Cones.map b\<guillemotright>"
these_not_empty_eq: fixes B :: "'b option set" shows "(Option.these B \<noteq> {}) = (B \<noteq> {} \<and> B \<noteq> {None})"
simps: fixes P :: "nat list \<Rightarrow> nat \<Rightarrow> bool"   and as :: "nat list"   and xs :: "nat list"   and s :: "nat" shows "suffs P as (xs, s) = (length xs = length as \<and> s = as \<bullet> xs \<and> (\<forall>i\<le>length xs. cond_cons P (drop i xs, drop i as \<bullet> drop i xs)))"
naturality: fixes f :: "'c" assumes "C.arr f" shows "C.map (C.cod f) \<cdot>\<^sub>C C.map f = C.map f \<cdot>\<^sub>C C.map (C.dom f)"
abs_le_self_iff: fixes abs :: "'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq (abs a) a = less_eq zero a"
arity_hd_ne_infinity_if_\<delta>_gt_0: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> 'c"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and \<zeta> :: "('b, 'a) hd" assumes "kbo_std_basis ground_heads_var gt_sym arity_sym arity_var wt_sym \<epsilon> \<delta> extf"   and "0 < \<delta>" shows "arity.arity_hd arity_sym arity_var \<zeta> \<noteq> \<infinity>"
until_induct: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<psi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list"   and \<chi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool" assumes "until \<phi> \<psi> \<pi>l"   and "\<And>\<pi>l. \<psi> \<pi>l \<Longrightarrow> \<chi> \<pi>l"   and "\<And>\<pi>l. \<lbrakk>\<phi> \<pi>l; until \<phi> \<psi> (map stl \<pi>l); \<chi> (map stl \<pi>l)\<rbrakk> \<Longrightarrow> \<chi> \<pi>l" shows "\<chi> \<pi>l"
SkipE: fixes s :: "char list \<Rightarrow> int"   and ct :: "com \<times> (char list \<Rightarrow> int)"   and P :: "bool" assumes "(SKIP, s) \<rightarrow> ct" shows "P"
crsp_step_dec: fixes ly :: "nat list"   and ap :: "abc_inst list"   and as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list"   and n :: "nat"   and e :: "nat" assumes "ly = layout_of ap"   and "crsp ly (as, lm) (s, l, r) ires"   and "abc_fetch as ap = Some (Dec n e)" shows "\<exists>stp>0. crsp ly (abc_step_l (as, lm) (Some (Dec n e))) (steps (s, l, r) (ci ly (start_of ly as) (Dec n e), start_of ly as - Suc 0) stp) ires"
Dom_fun_eq_set: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and \<UU>A :: "'e set"   and B :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and \<UU>B :: "'g set" assumes "Domainp A = (\<lambda>x. x \<in> \<UU>A)"   and "bi_unique A"   and "right_total A"   and "Domainp B = (\<lambda>x. x \<in> \<UU>B)"   and "bi_unique B"   and "right_total B" shows "{f. f ` \<UU>A \<subseteq> \<UU>B} = Collect (Domainp (A ===> B))"
map_replicate: fixes f :: "'d \<Rightarrow> 'c"   and n :: "nat"   and x :: "'d" shows "map f (replicate n x) = replicate n (f x)"
Radmit_def: fixes \<sigma> :: "('sf, 'sc, 'sz) subst"   and R :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list \<times> ('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list" shows "Radmit \<sigma> R = ((\<forall>i\<ge>0. i < length (fst R) \<longrightarrow> Sadmit \<sigma> (fst R ! i)) \<and> Sadmit \<sigma> (snd R))"
mkPX_def: fixes P :: "pol"   and i :: "nat"   and Q :: "pol" shows "mkPX P i Q = (case P of Pc c \<Rightarrow> if c = 0 then mkPinj 1 Q else PX P i Q | Pinj j R \<Rightarrow> PX P i Q | PX P2 i2 Q2 \<Rightarrow> if Q2 = Pc 0 then PX P2 (i + i2) Q else PX P i Q)"
direct_decomp_appendD_3: fixes A :: "'a set"   and ss1 :: "'a set list"   and ss2 :: "'a set list" assumes "direct_decomp A (ss1 @ ss2)" shows "direct_decomp A [sum_list ` listset ss1, sum_list ` listset ss2]"
one_star_lam_cong: fixes t1 :: "lam"   and t2 :: "lam"   and a :: "name" assumes " t1 \<longrightarrow>\<^sub>1\<^sup>* t2" shows " Lam [a].t1 \<longrightarrow>\<^sub>1\<^sup>* Lam [a].t2"
that: assumes "\<Z>_sgrp \<alpha> \<SS>"   and "\<And>a b. \<lbrakk>a \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>; b \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>\<rbrakk> \<Longrightarrow> a \<odot>\<^sub>\<circ>\<^bsub>\<SS>\<^esub> b = b \<odot>\<^sub>\<circ>\<^bsub>\<SS>\<^esub> a" shows "thesis"
"1_1": assumes "\<not> (x < 0 \<or> n < 0)"   and "\<not> x ^ p \<le> n"   and "0 \<le> (n div x ^ pm + x * int pm) div int p"   and "0 \<le> n" shows "root_main' ((n div x ^ pm + x * int pm) div int p) n = root_newton_int_main ((n div x ^ pm + x * int pm) div int p) n \<and> (root_main' ((n div x ^ pm + x * int pm) div int p) n = (y, b) \<longrightarrow> 0 \<le> y \<and> y ^ p \<le> n \<and> b = (y ^ p = n))"
simps_17: fixes x1 :: "('d, 'c) hd" shows "vars (Hd x1) = vars_hd x1"
set: fixes x1 :: "('a \<times> real) msllist"   and x2 :: "real \<Rightarrow> real" shows "set_ms (MS x1 x2) = \<Union> (Basic_BNFs.fsts ` set_msllist x1)"
char_one: fixes G :: "('c, 'd) monoid_scheme"   and \<chi> :: "'c \<Rightarrow> complex" assumes "character G \<chi>" shows "\<chi> \<one>\<^bsub>G\<^esub> = 1"
simps_19: fixes uy :: "'b"   and v :: "'b rexp" shows "le_rexp (Atom uy) (Star v) = True"
sum_list_map2_plus: fixes xs :: "'a list"   and ys :: "'a list" assumes "length xs = length ys" shows "sum_list (map2 (+) xs ys) = sum_list xs + sum_list ys"
locals_set_locals: fixes l :: "lname \<Rightarrow> val option"   and s :: "st" shows "locals (set_locals l s) = l"
hm_update_def: fixes k :: "'a"   and v :: "'b"   and hm :: "('a, 'b) hashmap" shows "hm_update k v hm = RBT_HM (HashMap_Impl.update k v (impl_of_RBT_HM hm))"
LEX_def: shows "Term_Order.LEX = Abs_nat_term_order lex_comp"
wens_weakening: fixes B :: "'b set"   and F :: "'b program"   and act :: "('b \<times> 'b) set" shows "B \<subseteq> wens F act B"
kernel_dim_code: fixes A :: "'c mat" shows "kernel_dim A = dim_col A - length (pivot_positions (gauss_jordan_single A))"
simps_3: fixes t :: "('c, 'd) SkewBinomialTree"   and t' :: "('c, 'd) SkewBinomialTree"   and bq :: "('c, 'd) SkewBinomialTree list" shows "rank_invar (t # t' # bq) = (rank t < rank t' \<and> rank_invar (t' # bq))"
disjoint_iff_not_equal: fixes A :: "'b set"   and B :: "'b set" shows "(A \<inter> B = {}) = (\<forall>x\<in>A. \<forall>y\<in>B. x \<noteq> y)"
c: shows "b \<otimes> inv (a \<otimes> b) \<in> carrier G"
prec_def: shows "prec \<equiv> \<lambda>f g. (pr_strict' f g, pr_weak' f g)"
smooth_bump_functionE: fixes A :: "'a set"   and U :: "'a set" assumes "closedin (top_of_set W'.src.carrier) A"   and "A \<subseteq> U"   and "U \<subseteq> W'.src.carrier"   and "open U" obtains \<psi> :: "'a \<Rightarrow> real" where "diff_fun k (charts_submanifold W') \<psi>" and "\<And>x. x \<in> W'.src.carrier \<Longrightarrow> 0 \<le> \<psi> x" and "\<And>x. x \<in> W'.src.carrier \<Longrightarrow> \<psi> x \<le> 1" and "\<And>x. x \<in> A \<Longrightarrow> \<psi> x = 1" and "csupport_on W'.src.carrier \<psi> \<inter> W'.src.carrier \<subseteq> U"
order_iff_strict: fixes b :: "'a \<Rightarrow>\<^sub>0 'c"   and a :: "'a \<Rightarrow>\<^sub>0 'c" shows "punit.ord_p b a = (punit.ord_strict_p b a \<or> a = b)"
eq_integralD: fixes k :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and y :: "'c" assumes "integral k f = y" shows "(f has_integral y) k \<or> \<not> f integrable_on k \<and> y = (0::'c)"
OO_def: fixes R :: "'b \<Rightarrow> 'e \<Rightarrow> bool"   and S :: "'e \<Rightarrow> 'd \<Rightarrow> bool" shows "R OO S = (\<lambda>x z. \<exists>y. R x y \<and> S y z)"
IN_FDOM_DRESTRICT_DIFF: fixes v :: "'d"   and vs :: "'d set"   and f :: "('d, 'e) fmap"   and fdom :: "'d set" assumes "v \<notin> vs"   and "fmdom' f \<subseteq> fdom"   and "v \<in> fmdom' f" shows "v \<in> fmdom' (fmrestrict_set (fdom - vs) f)"
wf_subset: fixes r :: "('d \<times> 'd) set"   and p :: "('d \<times> 'd) set" assumes "wf r"   and "p \<subseteq> r" shows "wf p"
le_fininf_suffixI: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w :: "'a stream"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a stream" assumes "traces ind"   and "traces.eq_inf ind w (w\<^sub>1 @- w\<^sub>2)" shows "traces.le_fininf ind w\<^sub>1 w"
set_of_Node: fixes lt :: "dag"   and a :: "ref"   and rt :: "dag" shows "set_of (Node lt a rt) = {a} \<union> set_of lt \<union> set_of rt"
assms_1: shows "Controller_L s (fin_inf_append [Zero] l) l z"
elims: fixes x :: "('a \<times> 'b) bound"   and y :: "nat" assumes "local.size x = y" obtains l0 :: "'a" and r0 :: "'b" and l1 :: "'a" and r1 :: "'b" where "x = Bound (l0, r0) (l1, r1)" and "y = size0 (Bound l0 l1) * size1 (Bound r0 r1)"
homeomorphic_maps_imp_map: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a" assumes "homeomorphic_maps X Y f g" shows "homeomorphic_map X Y f"
simps_1: shows "wpo s t = (case s of Var x \<Rightarrow> (False, case t of Var xa \<Rightarrow> x = xa | Fun g ts \<Rightarrow> (s, t) \<in> NS \<and> \<sigma> (g, length ts) = [] \<and> prl (g, length ts)) | Fun f ss \<Rightarrow> if (s, t) \<in> S then (True, True) else if (s, t) \<in> NS then if \<exists>i\<in>set (\<sigma> (f, length ss)). ss ! i \<succeq> t then (True, True) else case t of Var x \<Rightarrow> (False, ssimple \<and> large (f, length ss)) | Fun g ts \<Rightarrow> case prc (f, length ss) (g, length ts) of (prs, prns) \<Rightarrow> if prns \<and> (\<forall>j\<in>set (\<sigma> (g, length ts)). s \<succ> ts ! j) then if prs then (True, True) else let ss' = map ((!) ss) (\<sigma> (f, length ss)); ts' = map ((!) ts) (\<sigma> (g, length ts)); cf = c (f, length ss); cg = c (g, length ts) in if cf = Lex \<and> cg = Lex then lex_ext wpo n ss' ts' else if cf = Mul \<and> cg = Mul then mul_ext wpo ss' ts' else (length ss' \<noteq> 0 \<and> length ts' = 0, length ts' = 0) else (False, False) else (False, False))"
is_rbt_RBT_Impl_diag: fixes lt :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "('c, 'd) rbt"   and leq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "ord.is_rbt lt t" shows "ord.is_rbt (less_prod leq lt lt) (RBT_Impl_diag t)"
satisfies_tableau_def: fixes v :: "nat \<Rightarrow> 'b"   and t :: "(nat \<times> linear_poly) list" shows "v \<Turnstile>\<^sub>t t \<equiv> \<forall>e\<in>set t. v \<Turnstile>\<^sub>e e"
tt_ge: fixes v :: "'t"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "\<And>u. u \<prec>\<^sub>t v \<Longrightarrow> lookup p u = (0::'c)"   and "p \<noteq> 0" shows "v \<preceq>\<^sub>t tt p"
intros: fixes P :: "pi"   and Q :: "pi" assumes "P \<leadsto>[bisim] Q"   and "Q \<sim> P" shows "P \<sim> Q"
legal_min_zpassign: shows "legal_zpassign (\<lambda>v. zhmset_of (min_passign v))"
deadlock_free_is_deadlock_free_v2: fixes P :: "'d process" assumes "deadlock_free P" shows "deadlock_free_v2 P"
chain_boundary_simplicial_cone: fixes p :: "nat"   and S :: "(nat \<Rightarrow> real) set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int"   and v :: "nat \<Rightarrow> real" assumes "simplicial_chain p S c" shows "chain_boundary (Suc p) (simplicial_cone p v c) = c - (if p = 0 then frag_extend (\<lambda>f. frag_of (\<lambda>u\<in>standard_simplex p. v)) c else simplicial_cone (p - 1) v (chain_boundary p c))"
\<omega>_limit_set_positively_invariant: fixes x :: "'a" shows "rev.positively_invariant (rev.\<omega>_limit_set x)"
R: assumes "C = {#s#} + (C - {#s#})"   and "A = B + (C - {#s#})" shows "P"
tab_simps_1: shows "D.seq (D.inv (\<Phi> (r, f))) (F \<rho>)"
simps: fixes \<phi> :: "'d stream \<Rightarrow> bool"   and a :: "'d stream" shows "ev \<phi> a = ((\<exists>xs. a = xs \<and> \<phi> xs) \<or> (\<exists>xs. a = xs \<and> ev \<phi> (stl xs)))"
Process_eq_spec_optimized: shows "((P `;` Q) = (U `;` S)) = (D (P `;` Q) = D (U `;` S) \<and> (D (P `;` Q) = D (U `;` S) \<longrightarrow> F (P `;` Q) = F (U `;` S)))"
strict_order_greek_less: fixes r :: "('a \<times> 'a) set" assumes "strict_order r" shows "strict_order (greek_less r)"
S6Hist: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> HNext rmhist p \<and> $S6 rmhist p \<and> c p$ = $c p \<longrightarrow> (rmhist!p)$ = $(rmhist!p)"
seqI': fixes f :: "'c \<times> 'c"   and a :: "'c \<times> 'c"   and b :: "'c \<times> 'c"   and g :: "'c \<times> 'c"   and c :: "'c \<times> 'c" assumes "C.VV.in_hom f a b"   and "C.VV.in_hom g b c" shows "C.VV.seq g f"
D': shows "\<forall>a b c. ideal_generated {a, b, c} = ideal_generated {1::'a} \<longrightarrow> (\<exists>p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1::'a})"
open_ball_chartE: fixes charts :: "('c, 'd) chart set"   and k :: "enat"   and x :: "'c"   and U :: "'c set" assumes "c_manifold charts k"   and "x \<in> U"   and "open U"   and "U \<subseteq> manifold.carrier charts" obtains c :: "('c, 'd) chart" and r :: "real" where "c \<in> c_manifold.atlas charts k" and "x \<in> domain c" and "domain c \<subseteq> U" and "codomain c = ball (apply_chart c x) r" and "0 < r"
3: shows "sum f1 B1 = sum f1 A1 - f1 a1"
HSem_eq: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and \<rho> :: "Vars.var \<Rightarrow> Value" shows "\<lbrace>\<Gamma>\<rbrace>\<rho> = \<rho> ++\<^bsub>domA \<Gamma>\<^esub> \<^bold>\<lbrakk> \<Gamma> \<^bold>\<rbrakk>\<^bsub>\<lbrace>\<Gamma>\<rbrace>\<rho>\<^esub>"
bell_11_index_3: shows "|\<beta>\<^sub>1\<^sub>1\<rangle> $$ (2, 0) = complex_of_real (- 1 / sqrt 2)"
DJ: shows "ta_rstates TA \<inter> ta_rstates TA' = {}"
proots_within_def: fixes p :: "'a poly"   and s :: "'a set" shows "proots_within p s = {x \<in> s. poly p x = (0::'a)}"
on3: shows "\<forall>qs\<in>{x. set x \<subseteq> set init}. \<forall>(x, y)\<in>{(x, y). x \<in> set init \<and> y \<in> set init \<and> x < y}. T\<^sub>p_on_rand' A (fst A (Lxy init {x, y}) \<bind> (\<lambda>is. return_pmf (Lxy init {x, y}, is))) (Lxy qs {x, y}) \<le> c * real (T\<^sub>p_opt (Lxy init {x, y}) (Lxy qs {x, y})) + b"
axiom5_F: fixes f1 :: "'l \<Rightarrow> 'p"   and f2 :: "'m \<Rightarrow> 'o"   and f3 :: "'n \<Rightarrow> 'q" shows "F'set1 \<circ> F'map f2 f3 f1 = (`) f2 \<circ> F'set1"
generalized_sfw_simps_2: fixes a :: "'a simple_match \<times> 'b"   and as :: "('a simple_match \<times> 'b) list"   and p :: "('a, 'c) simple_packet_scheme" shows "generalized_sfw (a # as) p = (if case a of (m, x) \<Rightarrow> simple_matches m p then Some a else generalized_sfw as p)"
1: shows "\<not> (dim_row A = 0 \<or> dim_col A = 0)"
less_node_ptr_def: fixes x :: "(_) node_ptr"   and y :: "(_) node_ptr" shows "x < y \<equiv> x \<le> y \<and> \<not> y \<le> x"
st_approx_self: fixes x :: "real star" assumes "x \<in> HFinite" shows "st x \<approx> x"
utility_prefs_corresp: fixes f :: "real \<Rightarrow> real" assumes "monotone (\<lambda>x y. y < x) (\<lambda>x y. y < x) f" shows "\<forall>x\<in>carrier. \<forall>y\<in>carrier. x \<succeq>[relation] y = ((f \<circ> u) y \<le> (f \<circ> u) x)"
hyps_3: assumes "P h'"   and "final h''"   and "tame h''" shows "h' [filter ok \<circ> succs]\<rightarrow>* h''"
simps_4: fixes e1 :: "cexpr"   and e2 :: "cexpr" shows "free_vars <e1, e2>\<^sub>c = free_vars e1 \<union> free_vars e2"
init_obj_pred_def: shows "init_obj_pred \<equiv> \<lambda>x xa. pred.Pred (init_obj' x xa)"
inter_Set_filter: fixes B :: "'b set"   and A :: "'b set" assumes "finite B" shows "A \<inter> B = Set.filter (\<lambda>x. x \<in> A) B"
domSubset4: fixes x :: "(int \<times> int) set set"   and y :: "(int \<times> int) set set"   and dn :: "int" shows "dom (C (DenyAllFromTo x y \<oplus> DenyAllFromTo y x \<oplus> AllowPortFromTo x y dn)) = dom (C (DenyAllFromTo x y \<oplus> DenyAllFromTo y x))"
coprimeI: fixes p :: "'a"   and q :: "'a" assumes "\<And>r. \<lbrakk>r dvd p; r dvd q\<rbrakk> \<Longrightarrow> r dvd (1::'a)" shows "coprime p q"
floor_eq4: fixes n :: "nat"   and x :: "real" assumes "real n \<le> x"   and "x < real (Suc n)" shows "nat \<lfloor>x\<rfloor> = n"
is_projection_on_iff_orthog: fixes M :: "'a set"   and \<pi> :: "'a \<Rightarrow> 'a" assumes "closed_csubspace M" shows "is_projection_on \<pi> M = (\<forall>h. h - \<pi> h \<in> orthogonal_complement M \<and> \<pi> h \<in> M)"
encL: fixes S :: "'procS"   and SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" shows "TargetTerm (\<lbrakk>S\<rbrakk>) \<R>\<lbrakk>\<cdot>\<rbrakk>L<SRel,TRel> SourceTerm S"
welltyped_Abss: fixes b :: "expr"   and U :: "Unary_PCF.type"   and T :: "Unary_PCF.type"   and xTs :: "(char list \<times> Unary_PCF.type) list" assumes "b ::: U"   and "T = map snd xTs \<rightarrow>\<rightarrow> U" shows "\<Lambda>[xTs] b ::: T"
complex_cnj_cnj: fixes z :: "complex" shows "cnj (cnj z) = z"
approx_add_right_iff: fixes b :: "'a star"   and a :: "'a star"   and c :: "'a star" shows "(b + a \<approx> c + a) = (b \<approx> c)"
eqvt: fixes p :: "perm"   and a :: "assn" shows "p \<bullet> asToHeap a = asToHeap (p \<bullet> a)"
disc_1: fixes xs :: "'c llist"   and ys :: "'d llist" assumes "lnull xs \<or> lnull ys" shows "lnull (lzip xs ys)"
prems_5: fixes \<rho> :: "complex mat" assumes "\<rho> \<in> carrier_mat m m \<and> partial_density_operator \<rho>" shows "partial_density_operator (DS \<rho>)"
comp_null_1: fixes f :: "'b" shows "B.null \<cdot>\<^sub>B f = B.null"
finite_induct: fixes F :: "'g set"   and P :: "'g set \<Rightarrow> bool" assumes "finite F"   and "P {}"   and "\<And>x F. \<lbrakk>finite F; x \<notin> F; P F\<rbrakk> \<Longrightarrow> P (insert x F)" shows "P F"
UP_car_memI: fixes R :: "('c, 'd) ring_scheme"   and k :: "nat"   and p :: "nat \<Rightarrow> 'c" assumes "UP_ring R"   and "\<And>n. k < n \<Longrightarrow> p n = \<zero>\<^bsub>R\<^esub>"   and "\<And>n. p n \<in> carrier R" shows "p \<in> carrier (UP R)"
lcs_mem_ldiv: fixes H :: "'a set"   and a :: "'a"   and b :: "'a" assumes "G \<guillemotright> H "   and "a \<in> carrier G"   and "b \<in> carrier G" shows "(a \<in> b \<diamondsuit> H) = (\<rho> b \<cdot> a \<in> H)"
distinct_13: fixes x21 :: "nat"   and x22 :: "nat"   and x23 :: "nat"   and x24 :: "nat"   and x25 :: "nat"   and x51 :: "nat"   and x52 :: "nat"   and x53 :: "nat" shows "Rrep x21 x22 x23 x24 x25 \<noteq> Pkt x51 x52 x53"
transitive: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" assumes "P \<simeq> Q"   and "Q \<simeq> R" shows "P \<simeq> R"
ts_process_traces: fixes T :: "'c list set" assumes "trace_set T" shows "traces (ts_process T) = T"
irreducible_primitive_connect: fixes f :: "'a poly" assumes "primitive f" shows "irreducible\<^sub>d f = irreducible f"
spmf_map_inj': fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c spmf"   and x :: "'c" assumes "inj f" shows "spmf (map_spmf f M) (f x) = spmf M x"
Cons_2: shows "(s, x # xs) \<in> seq ars"
k_def: shows "k \<equiv> \<lfloor>(\<theta> + pi) / (2 * pi)\<rfloor>"
rep_eq: fixes x :: "\<o>" shows "eval\<o> (\<^bold>\<lambda>\<^sup>0 x) = eval\<o> x"
neutral_const: fixes A :: "'d set" shows "aggregation.sum_0 (\<lambda>_. zero) A = zero"
wellformed_transaction_cases_1: fixes T :: "('e, 'f, 'g, 'h) prot_transaction"   and l :: "'h strand_label"   and x :: "(('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) stateful_strand_step" assumes "wellformed_transaction T"   and "(l, x) \<in> set (transaction_receive T)" shows "\<exists>t. x = receive\<langle>t\<rangle>"
assms_3: fixes x :: "'b"   and i :: "'a" assumes "x \<in> space M"   and "P i x" shows "i \<in> I"
section_retraction_of_iso_2: fixes f :: "'c"   and g :: "'c" assumes "C.iso f"   and "C.ide (f \<cdot>\<^sub>C g)" shows "C.inverse_arrows f g"
is_lb_empty: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and l :: "'c" assumes "class.po below" shows "po.is_lb below {} l"
assms_2: fixes i :: "V" assumes "i \<in>\<^sub>\<circ> I" shows "\<phi> i = \<pi>\<^sub>D\<^sub>G I \<AA> i \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF>"
LTL_U_one_step_unfolding: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltl"   and \<psi> :: "'a ltl" shows "w \<Turnstile> \<phi> U \<psi> = (w \<Turnstile> \<psi> \<or> w \<Turnstile> \<phi> \<and> w \<Turnstile> X (\<phi> U \<psi>))"
reverse_subst_lam: fixes f :: "(val \<times> val) fset"   and x :: "nat"   and e :: "exp"   and \<rho> :: "(nat \<times> val) list"   and v :: "exp"   and y :: "nat"   and e' :: "exp" assumes "VFun f \<in> E (ELam x e) \<rho>"   and "is_val v"   and "ELam x e = ELam x (subst y v e')"   and "x \<noteq> y"   and "\<forall>v1 v2. v2 \<in> E (subst y v e') ((x, v1) # \<rho>) \<longrightarrow> (\<exists>\<rho>' v'. v' \<in> E v [] \<and> v2 \<in> E e' \<rho>' \<and> \<rho>' \<approx> (y, v') # (x, v1) # \<rho>)" shows "\<exists>\<rho>' v''. v'' \<in> E v [] \<and> VFun f \<in> E (ELam x e') \<rho>' \<and> \<rho>' \<approx> (y, v'') # \<rho>"
domrangefix: fixes x :: "('a \<times> 'a) set" shows "(rdom x = x) = (rel_antirange_kleene_algebra.ars_r x = x)"
res: shows "LLL_implementation.LLL_many_reconstruction p (p ^ find_exponent p (sqrt_int_ceiling (2 ^ (5 * (degree f div 2) * (degree f div 2)) * \<parallel>f\<parallel>\<^sup>2 ^ (2 * (degree f div 2))))) f (hensel_lifting p (find_exponent p (sqrt_int_ceiling (2 ^ (5 * (degree f div 2) * (degree f div 2)) * \<parallel>f\<parallel>\<^sup>2 ^ (2 * (degree f div 2))))) f gs) = gs"
strict_partial_order_acyclic: fixes r :: "('b \<times> 'b) set" assumes "strict_partial_order r" shows "acyclic r"
leftmost_unaffected_Derives1: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and j :: "nat"   and a :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and b :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "CFG.leftmost \<NN> \<TT> j a"   and "j < i"   and "CFG.Derives1 \<NN> \<TT> \<RR> a i r b" shows "CFG.leftmost \<NN> \<TT> j b"
inject: fixes x2 :: "name"   and x3 :: "trm"   and x1 :: "stack"   and y2 :: "name"   and y3 :: "trm"   and y1 :: "stack" shows "([x2]x3\<ggreater>x1 = [y2]y3\<ggreater>y1) = ([x2].x3 = [y2].y3 \<and> x1 = y1)"
preserves_arr: fixes f :: "'a" assumes "Right_a.arr f" shows "Right_a.arr (Right_a.R f)"
if_false_symeval: fixes \<Gamma> :: "'a \<Rightarrow> 'a"   and b :: "'a upred"   and P :: "'a hrel"   and Q :: "'a hrel" assumes "\<Gamma> \<dagger> b = false" shows "\<Gamma> \<Turnstile> (P \<triangleleft> b \<triangleright>\<^sub>r Q) = \<Gamma> \<Turnstile> Q"
CLDomain_def: fixes T :: "(ZF, ZF) Axioms" shows "CLDomain T \<equiv> TDomain \<circ> ZF2m T"