hcju/isabelleps huggingface dataset:
r: fixes x :: "'s" assumes "x \<in> X" shows "measure_pmf.prob (K x) X \<le> r"
ins_node_def: fixes v :: "'b"   and g :: "'b ugraph" shows "ins_node v g \<equiv> graph (insert v (nodes g)) (edges g)"
STAR_star_of_Int: fixes X :: "'b set" shows "*s* X \<inter> Standard = star_of ` X"
simps: fixes a1 :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> Arity \<times> Arity list"   and a2 :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" shows "consistent a1 a2 = (\<exists>ae a as \<Gamma> e S. a1 = (ae, a, as) \<and> a2 = (\<Gamma>, e, S) \<and> a_consistent (ae, a, as) (\<Gamma>, e, S) \<and> (\<forall>x. x \<in> thunks \<Gamma> \<longrightarrow> ae x = up\<cdot>0))"
span_image_scale: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set"   and c :: "'b \<Rightarrow> 'a" assumes "vector_space scale"   and "finite S"   and "\<And>x. x \<in> S \<Longrightarrow> c x \<noteq> (0::'a)" shows "module.span scale ((\<lambda>x. scale (c x) x) ` S) = module.span scale S"
statesDecls_def: fixes s :: "(integer \<times> edge list) iarray" shows "statesDecls s = \<Union> (edgeDecls ` edgeSet s)"
seq_UINF_distl: fixes P :: "('b, 'd) urel"   and A :: "'e set"   and F :: "'e \<Rightarrow> ('d, 'c) urel" shows "P ;; (\<Sqinter> Q \<in> A \<bullet> F Q) = (\<Sqinter> Q \<in> A \<bullet> P ;; F Q)"
midpoint_def: fixes a :: "'a"   and b :: "'a" shows "midpoint a b = (a + b) /\<^sub>R 2"
degree_smult_eq: fixes a :: "'a"   and p :: "'a poly" shows "degree (smult a p) = (if a = (0::'a) then 0 else degree p)"
strict_trans2: fixes a :: "'a list"   and b :: "'a list"   and c :: "'a list" assumes "a <lex b"   and "b \<le>lex c" shows "a <lex c"
homotopic_loops_imp_path: fixes s :: "'a set"   and p :: "real \<Rightarrow> 'a"   and q :: "real \<Rightarrow> 'a" assumes "homotopic_loops s p q" shows "path p \<and> path q"
that: fixes n :: "nat" assumes "Lyndon (rotate n w)" shows "thesis"
card_Un_Int: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "finite B" shows "card A + card B = card (A \<union> B) + card (A \<inter> B)"
intro: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "compatible_ordering A less_eq less"   and "partially_ordered_set A less_eq" shows "partial_ordering A less_eq less"
integral: shows "((-) (t + h) has_integral h\<^sup>2 / 2) (cbox t (t + h))"
from_nat_inj: fixes i :: "nat"   and j :: "nat" assumes "i < CARD('d)"   and "j < CARD('d)"   and "Bij_Nat.from_nat i = Bij_Nat.from_nat j" shows "i = j"
list_all2_Cons: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list"   and y :: "'c"   and ys :: "'c list" shows "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
Val_abs1_axioms: shows "Val_abs1 num_ivl plus_ivl \<gamma>_ivl in_ivl filter_plus_ivl filter_less_ivl"
caratheodory_aff_dim: fixes p :: "'a set" shows "convex hull p = {x. \<exists>S. finite S \<and> S \<subseteq> p \<and> int (card S) \<le> aff_dim p + 1 \<and> x \<in> convex hull S}"
finite_regular: fixes A :: "'b list set" assumes "finite A" shows "regular A"
root_in_edge_sources: fixes r :: "'a"   and x :: "'a" assumes "connected_root r x"   and "x \<noteq> 0"   and "is_vector r" shows "r \<le> x ; 1"
lb_ln2_def: fixes prec :: "nat" shows "lb_ln2 prec = (let third = lapprox_rat prec 1 3 in float_plus_down prec (Float 1 (- 1) * lb_ln_horner prec (get_even prec) 1 (Float 1 (- 1))) (float_round_down prec (third * lb_ln_horner prec (get_even prec) 1 third)))"
least_squares_approximation4': fixes S :: "'a set"   and v :: "'a" assumes "subspace S" shows "\<exists>!p. p \<in> S \<and> (\<forall>y\<in>S. norm (v - p) \<le> norm (v - y))"
nonzero_memE_2: fixes a :: "nat \<Rightarrow> int" assumes "a \<in> nonzero Zp" shows "a \<noteq> \<zero>"
IH: fixes r :: "real" assumes "0 < r" shows "emeasure (Pi\<^sub>M A (\<lambda>_. lborel)) ({f. sqrt (\<Sum>i\<in>A. (f i)\<^sup>2) \<le> r} \<inter> space (Pi\<^sub>M A (\<lambda>_. lborel))) = ennreal (unit_ball_vol (real (card A)) * r ^ card A)"
conjugation_of_cycle: fixes cs :: "'a list"   and p :: "'a \<Rightarrow> 'b" assumes "cycle cs"   and "bij p" shows "p \<circ> cycle_of_list cs \<circ> inv' p = cycle_of_list (map p cs)"
assms: shows "b \<in>\<^sub>\<circ> r `\<^sub>\<circ> A"
iteration_mono_less: fixes x :: "int"   and n :: "int" assumes "0 \<le> x"   and "0 \<le> n"   and "n < x ^ p" shows "(n div x ^ pm + x * int pm) div int p < x"
relevant_props: fixes p :: "fm"   and i1 :: "char list \<Rightarrow> tv"   and i2 :: "char list \<Rightarrow> tv" assumes "\<forall>s\<in>props p. i1 s = i2 s" shows "eval i1 p = eval i2 p"
elims_2: fixes x :: "'b graph"   and xa :: "'b \<Rightarrow> nat" assumes "SINVAR_BLPbasic.sinvar x xa" obtains G :: "'b graph" and nP :: "'b \<Rightarrow> nat" where "x = G" and "xa = nP" and "\<forall>(e1, e2)\<in>edges G. nP e1 \<le> nP e2"
inf_llist: fixes f :: "nat \<Rightarrow> 'a" assumes "xs = inf_llist f" shows "thesis"
SetMem_strict: fixes x :: "nat discr Box" shows "SetMem\<cdot>x\<cdot>\<bottom> = \<bottom>"
Union_insert: fixes x :: "'a set"   and S :: "'a set set" shows "\<Union> (insert x S) = x \<union> \<Union> S"
neg_def: fixes S :: "'a set"   and s0 :: "'a"   and \<delta> :: "'a \<Rightarrow> 'a set"   and AP :: "'b set"   and L :: "'a \<Rightarrow> 'b set"   and AP' :: "'b set"   and \<phi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool"   and \<pi>l :: "('a \<times> 'b set) stream list" assumes "Shallow S s0 \<delta> AP L AP'" shows "Shallow.neg \<phi> \<pi>l \<equiv> \<not> \<phi> \<pi>l"
curry_in_hom: fixes f1 :: "'a1"   and F :: "'a1 \<times> 'a2 \<Rightarrow> 'b"   and G :: "'a1 \<times> 'a2 \<Rightarrow> 'b"   and \<tau> :: "'a1 \<times> 'a2 \<Rightarrow> 'b" assumes "A1.arr f1"   and "natural_transformation (\<cdot>\<^sub>A\<^sub>1\<^sub>x\<^sub>A\<^sub>2) (\<cdot>\<^sub>B) F G \<tau>" shows "\<guillemotleft>local.curry F G \<tau> f1 : local.curry F F F (A1.dom f1) \<rightarrow>\<^sub>[\<^sub>A\<^sub>2\<^sub>,\<^sub>B\<^sub>] local.curry G G G (A1.cod f1)\<guillemotright>"
runit'_simps_3: fixes f :: "'a" assumes "ide f" shows "trg \<r>\<^sup>-\<^sup>1[f] = trg f"
Rep_comp_hom: fixes f :: "'b nd_fun"   and g :: "'b nd_fun" shows "Rep_nd_fun (f \<cdot> g) = Rep_nd_fun f \<circ>\<^sub>K Rep_nd_fun g"
neg_lit_in_atms_of: fixes A :: "'a"   and C :: "'a literal multiset" assumes "Neg A \<in># C" shows "A \<in> atms_of C"
assms_2: shows "\<pi>2 permutes {..<length ys}"
Send_L1: fixes return :: "nat \<Rightarrow> 'a Frame list"   and send :: "nat \<Rightarrow> 'a Frame list"   and get :: "nat \<Rightarrow> nat list"   and activation :: "nat \<Rightarrow> nat list"   and t :: "nat" assumes "Send return send get activation"   and "send t \<noteq> []" shows "activation t \<noteq> []"
map_option_eq_Some: fixes f :: "'d \<Rightarrow> 'c"   and xo :: "'d option"   and y :: "'c" shows "(map_option f xo = Some y) = (\<exists>z. xo = Some z \<and> f z = y)"
setdist_eq_0_sing_2: fixes S :: "'b set"   and x :: "'b" shows "(setdist S {x} = 0) = (S = {} \<or> x \<in> closure S)"
ballg: fixes z :: "complex" assumes "z \<in> ball 0 r" shows "g z \<in> ball 0 r"
is_extensional: fixes f :: "'a arrow_of_spans_data" assumes "\<not> arr f" shows "trg f = null"
l: shows "left x \<le> s \<and> s \<le> right x"
collapse: fixes prod :: "'l \<times> 'm" shows "(fst prod, snd prod) = prod"
assms: fixes x :: "'b"   and y :: "'b" shows "f (x + y) = f x + f y"
1: shows "2 * x * y \<le> (x + y - 1)\<^sup>2"
deriv_R: shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c R,B"
snd_vimage_eq_Times: fixes S :: "'b set" shows "snd -` S = UNIV \<times> S"
simps_6: fixes t1 :: "Formula.trm"   and t2 :: "Formula.trm" shows "formula_of_constraint (t1, False, MLessEq, t2) = formula.Neg (formula.LessEq t1 t2)"
skolemize1: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "\<forall>x. P x \<longrightarrow> (\<exists>y. Q x y)" shows "\<exists>f. \<forall>x. P x \<longrightarrow> Q x (f x)"
pre_oldest_configuration_tokens: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and q :: "'d"   and n :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w" shows "(semi_mojmir.pre_oldest_tokens \<delta> q\<^sub>0 w q n \<noteq> {}) = (semi_mojmir_def.configuration \<delta> q\<^sub>0 w q (Suc n) \<noteq> {})"
swap_simps_3: fixes a :: "atom"   and c :: "atom"   and b :: "atom" assumes "a \<noteq> c"   and "b \<noteq> c" shows "[(a, b)] \<bullet> c = c"
and_neq_0_is_nth: fixes y :: "'b word"   and n :: "nat"   and x :: "'b word" assumes "y = 2 ^ n" shows "(x AND y \<noteq> 0) = x !! n"
forward_match_final: fixes s1 :: "('fenv_std, 'henv, ('fun, 'dyn) frame) state"   and s2 :: "('fenv_inca, 'henv, ('fun, 'dyn) frame) state" assumes "s1 \<sim> s2"   and "Sstd.final s1" shows "Sinca.final s2"
bind_output_cong: fixes \<G> :: "'a cyclic_group"   and x :: "nat"   and b :: "nat"   and ab :: "nat"   and aa :: "nat"   and ac :: "nat" assumes "pedersen \<G>"   and "x < order \<G>" shows "(x = nat ((int b - int ab) * Number_Theory_Aux.inverse (aa - ac) (order \<G>) mod int (order \<G>))) = [x = nat ((int b - int ab) * Number_Theory_Aux.inverse (aa - ac) (order \<G>) mod int (order \<G>))] (mod order \<G>)"
linorder_on_linorder_from_keys: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "inj_on f A" shows "linorder_on A (linorder_from_keys A f)"
dirichlet_prod_commutes: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" shows "dirichlet_prod f g = dirichlet_prod g f"
Point': shows "p\<^sub>1 = Point x\<^sub>1 y\<^sub>1"
init_yields_initd: fixes G :: "prog"   and s1 :: "st"   and C :: "qtname"   and s2 :: "state" assumes "G\<turnstile>Norm s1 \<midarrow>Init C\<rightarrow> s2" shows "initd C s2"
roundup_nat_ge_8: fixes M :: "bit list"   and emBits :: "nat" assumes "length (sha1 M) + sLen + 16 \<le> emBits" shows "8 \<le> roundup emBits 8 * 8 - (length (sha1 M) + 8)"
add_minus_id: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c"   and p\<^sub>1 :: "'c point"   and p\<^sub>2 :: "'c point" assumes "ell_field R"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "ell_field.nonsingular R a b"   and "ell_field.on_curve R a b p\<^sub>1"   and "ell_field.on_curve R a b p\<^sub>2" shows "ell_field.add R a (ell_field.add R a p\<^sub>1 p\<^sub>2) (ell_field.opp R p\<^sub>2) = p\<^sub>1"
lookup_union2: fixes k :: "'a"   and ys :: "'a \<rightharpoonup>\<^sub>f 'b"   and xs :: "'a \<rightharpoonup>\<^sub>f 'b" assumes "k |\<notin>| fmdom ys" shows "lookup (xs ++\<^sub>f ys) k = lookup xs k"
hnr_RETURN_pure: fixes c :: "'a"   and a :: "'b"   and R :: "('a \<times> 'b) set" assumes "(c, a) \<in> R" shows "hn_refine emp (return c) emp (pure R) (RETURN a)"
prod_emb_INT: fixes I :: "'b set"   and M :: "'c set"   and L :: "'c \<Rightarrow> 'd measure"   and K :: "'c set"   and F :: "'b \<Rightarrow> ('c \<Rightarrow> 'd) set" assumes "I \<noteq> {}" shows "prod_emb M L K (\<Inter> (F ` I)) = (\<Inter>i\<in>I. prod_emb M L K (F i))"
intro: fixes \<Omega> :: "'c set"   and M :: "'c set set" assumes "algebra \<Omega> M"   and "sigma_algebra_axioms M" shows "sigma_algebra \<Omega> M"
SG_dec: shows "(\<Gamma>, \<Delta>) = SG ! i"
hv_zk2: fixes H :: "'a \<times> 'a"   and w :: "nat"   and c :: "nat" assumes "(H, w) \<in> R" shows "chaum_ped_sigma.R H w c = chaum_ped_sigma.S H c"
normalize_gauss_int_of_numeral: fixes n' :: "num" shows "normalize (numeral n') = numeral n'"
runit_simps_2: fixes f :: "'a" assumes "B.ide f" shows "src\<^sub>B (B.runit f) = src\<^sub>B f"
prim_invar_impl_def: fixes Qi :: "'q"   and \<pi>i :: "'m" shows "prim_invar_impl Qi \<pi>i \<equiv> Q_invar Qi \<and> M_invar \<pi>i \<and> prim_invar2 (Q_\<alpha> Qi) (M_lookup \<pi>i)"
converseE: fixes yx :: "'e \<times> 'f"   and r :: "('f \<times> 'e) set" assumes "yx \<in> r\<inverse>" obtains x :: "'f" and y :: "'e" where "yx = (y, x)" and "(x, y) \<in> r"
split: fixes P :: "'c \<Rightarrow> bool"   and f :: "'d match_expr \<Rightarrow> action \<Rightarrow> 'c"   and rule :: "'d rule" shows "P (case rule of Rule x xa \<Rightarrow> f x xa) = (\<forall>x1 x2. rule = Rule x1 x2 \<longrightarrow> P (f x1 x2))"
per_init_of_nat_range: fixes N :: "nat" shows "per_init {i. i < N} = per_init' N"
negate_eq2: fixes x :: "'a" assumes "x \<in> V" shows "- 1 \<cdot> x = - x"
distrib_right': fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.ab_near_semiring plus times" shows "times (plus x y) z = plus (times x z) (times y z)"
succ_it_def: fixes g :: "'G"   and v :: "'V" shows "succ_it g v \<equiv> it_to_it (succ_list_it g v)"
app: fixes t :: "'c"   and u :: "'c"   and S :: "name fset" shows "closed_except (app t u) S = (closed_except t S \<and> closed_except u S)"
inf_sup_ord_4: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and y :: "'c"   and x :: "'c" assumes "class.lattice inf less_eq less sup" shows "less_eq y (sup x y)"
f2: shows "Deliver (i, Create i e) \<in> set pre \<or> Deliver (i, Append i e) \<in> set pre \<or> (\<exists>mo. Deliver (i, Expunge e mo i) \<in> set pre) \<or> (\<exists>mo. Deliver (i, Store e mo i) \<in> set pre)"
left_inverse: fixes a :: "'c" shows "- a + a = (0::'c)"
assms_7: shows "integrable M (\<lambda>x. \<bar>g x\<bar> powr q)"
ocrit_specI: fixes crit :: "nat \<times> 'e \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (bool \<times> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g \<Rightarrow> bool" assumes "\<And>d m data hs ps B p q. \<lbrakk>dickson_grading d; set hs \<subseteq> B; fst ` B \<subseteq> dgrad_p_set d m; unique_idx (p # q # hs @ map (fst \<circ> snd) ps @ map (snd \<circ> snd) ps) data; \<And>p' q'. \<lbrakk>(p', q') \<in>\<^sub>p snd ` set ps; fst p' \<noteq> 0; fst q' \<noteq> 0\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst p') (fst q'); p \<in> B; q \<in> B; fst p \<noteq> 0; fst q \<noteq> 0; crit data hs ps p q\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst p) (fst q)" shows "ocrit_spec crit"
simple_function_const: fixes M :: "'b measure"   and c :: "'c" shows "simple_function M (\<lambda>x. c)"
simps_3: fixes x :: "'b"   and xs :: "['b]"   and y :: "'b"   and ys :: "['b]" shows "isPrefixOf\<cdot>(x : xs)\<cdot>(y : ys) = (eq\<cdot>x\<cdot>y andalso isPrefixOf\<cdot>xs\<cdot>ys)"
pps_to_list_def: fixes S :: "('x option \<Rightarrow>\<^sub>0 nat) set" shows "pp_pm.extended_ord.punit.pps_to_list S = rev (pp_pm.extended_ord.ordered_powerprod_lin.sorted_list_of_set S)"
homeomorphism_on_extension_sequentially: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and T :: "'c set" assumes "\<And>u b. \<forall>n. u n \<in> S \<Longrightarrow> u \<longlonglongrightarrow> b = (\<lambda>n. f (u n)) \<longlonglongrightarrow> f b"   and "T \<subseteq> closure S" shows "homeomorphism_on T f"
simplicialcomplex_image: shows "SimplicialComplex (f \<turnstile> X)"
normal_drop: fixes ns :: "nat list" shows "normal (dropWhile (\<lambda>n. n = 0) ns)"
complete_lattice_axioms_def: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and bot :: "'a"   and top :: "'a" shows "class.complete_lattice_axioms Inf Sup less_eq bot top \<equiv> ((\<forall>x A. x \<in> A \<longrightarrow> less_eq (Inf A) x) \<and> (\<forall>A z. (\<forall>x. x \<in> A \<longrightarrow> less_eq z x) \<longrightarrow> less_eq z (Inf A)) \<and> (\<forall>x A. x \<in> A \<longrightarrow> less_eq x (Sup A))) \<and> (\<forall>A z. (\<forall>x. x \<in> A \<longrightarrow> less_eq x z) \<longrightarrow> less_eq (Sup A) z) \<and> Inf {} = top \<and> Sup {} = bot"
bounded: shows "dbm_entry_val u (Some c) (Some c) (len M i i xs)"
fst_notin_fmap_minus_dom: fixes x :: "'g \<times> 'h"   and F :: "('g \<times> 'h) set"   and F' :: "'g \<Rightarrow> 'h option" assumes "insert x F = set_fmap F'" shows "fst x \<notin> dom (F' -- x)"
CARD_code: shows "CARD('a) = of_phantom Cardinality.card_UNIV'"
zero_im_sheaf_def: shows "zero_im_sheaf \<equiv> \<lambda>V. zero_str\<^sub>X (f \<^sup>\<inverse> X V)"
simps_8: fixes v :: "'b poly"   and va :: "nat" shows "isnpolyh (Pw v va) = (\<lambda>k. False)"
equiv_utrans_le_utrans: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and u :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and v :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "equiv_utrans t u"   and "le_utrans u v" shows "le_utrans t v"
Dom_comp: fixes f :: "('obj, 'obj) concrete_category.arr"   and g :: "('obj, 'obj) concrete_category.arr" assumes "arr f"   and "arr g"   and "Dom g = Cod f" shows "Dom (g \<cdot> f) = Dom f"
derivs_One: fixes xs :: "'b list" shows "derivs xs One \<in> {Zero, One}"
T_quasi_preserves_null2_2: fixes A :: "'a set"   and n :: "nat" assumes "A \<in> null_sets M" shows "(T ^^ n) --` A \<in> null_sets M"
expand: fixes pr_operation' :: "pr_operation'"   and pr_operation'' :: "pr_operation'" assumes "is_RELABEL' pr_operation' = is_RELABEL' pr_operation''"   and "is_NONSAT_PUSH' pr_operation' = is_NONSAT_PUSH' pr_operation''"   and "\<lbrakk>is_SAT_PUSH' pr_operation'; is_SAT_PUSH' pr_operation''\<rbrakk> \<Longrightarrow> un_SAT_PUSH' pr_operation' = un_SAT_PUSH' pr_operation''" shows "pr_operation' = pr_operation''"
FOREACHci_refine: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and S' :: "'d set"   and \<sigma>0 :: "'e"   and \<sigma>0' :: "'f"   and R :: "('e \<times> 'f) set"   and \<Phi>'' :: "'c set \<Rightarrow> 'e \<Rightarrow> 'd set \<Rightarrow> 'f \<Rightarrow> bool"   and \<Phi>' :: "'d set \<Rightarrow> 'f \<Rightarrow> bool"   and \<Phi> :: "'c set \<Rightarrow> 'e \<Rightarrow> bool"   and c :: "'e \<Rightarrow> bool"   and c' :: "'f \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e \<Rightarrow> 'e nres"   and f' :: "'d \<Rightarrow> 'f \<Rightarrow> 'f nres" assumes "inj_on \<alpha> S"   and "S' = \<alpha> ` S"   and "(\<sigma>0, \<sigma>0') \<in> R"   and "\<Phi>'' S \<sigma>0 (\<alpha> ` S) \<sigma>0'"   and "\<And>it \<sigma> it' \<sigma>'. \<lbrakk>it' = \<alpha> ` it; it \<subseteq> S; it' \<subseteq> S'; \<Phi>' it' \<sigma>'; \<Phi>'' it \<sigma> it' \<sigma>'; \<Phi> it \<sigma>; (\<sigma>, \<sigma>') \<in> R\<rbrakk> \<Longrightarrow> c \<sigma> = c' \<sigma>'"   and "\<And>it \<sigma> it' \<sigma>'. \<lbrakk>it' = \<alpha> ` it; it \<subseteq> S; it' \<subseteq> S'; \<Phi>' it' \<sigma>'; \<Phi>'' it \<sigma> it' \<sigma>'; (\<sigma>, \<sigma>') \<in> R\<rbrakk> \<Longrightarrow> \<Phi> it \<sigma>"   and "\<And>x it \<sigma> x' it' \<sigma>'. \<lbrakk>x' = \<alpha> x; x \<in> it; x' \<in> it'; it' = \<alpha> ` it; it \<subseteq> S; it' \<subseteq> S'; \<Phi> it \<sigma>; \<Phi>' it' \<sigma>'; \<Phi>'' it \<sigma> it' \<sigma>'; c \<sigma>; c' \<sigma>'; (\<sigma>, \<sigma>') \<in> R\<rbrakk> \<Longrightarrow> f x \<sigma> \<le> \<Down> {(\<sigma>, \<sigma>'). (\<sigma>, \<sigma>') \<in> R \<and> \<Phi>'' (it - {x}) \<sigma> (it' - {x'}) \<sigma>'} (f' x' \<sigma>')" shows "FOREACH\<^sub>C\<^bsup>\<Phi>\<^esup> S c f \<sigma>0 \<le> \<Down> R (FOREACH\<^sub>C\<^bsup>\<Phi>'\<^esup> S' c' f' \<sigma>0')"
polysubst0_I: fixes b :: "'a"   and bs :: "'a list"   and a :: "poly"   and t :: "poly" shows "\<lparr>polysubst0 a t\<rparr>\<^sub>p\<^bsup>b # bs\<^esup> = \<lparr>t\<rparr>\<^sub>p\<^bsup>\<lparr>a\<rparr>\<^sub>p\<^bsup>b # bs\<^esup> # bs\<^esup>"
t_combine_eq_time_combine_tm: fixes p\<^sub>0\<^sub>L :: "int \<times> int"   and p\<^sub>1\<^sub>L :: "int \<times> int"   and p\<^sub>0\<^sub>R :: "int \<times> int"   and p\<^sub>1\<^sub>R :: "int \<times> int"   and l :: "int"   and ps :: "(int \<times> int) list" shows "t_combine (p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) (p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) l ps = time (combine_tm (p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) (p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) l ps)"
assms_1: shows " t1 \<longrightarrow>\<^sub>\<beta>\<^sup>* t2"
indep_inI: fixes X :: "'a set"   and \<E> :: "'a set" assumes "X \<subseteq> \<E>"   and "indep_in carrier X" shows "\<E>.indep_in \<E> X"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_Cons_3: fixes l :: "'d strand_label"   and a :: "poscheckvariant"   and t :: "('e, 'f) Term.term"   and s :: "('e, 'f) Term.term"   and A :: "('d strand_label \<times> ('e, 'f) stateful_strand_step) list" shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t ((l, \<langle>a: t \<doteq> s\<rangle>) # A) = (l, \<langle>a: t \<doteq> s\<rangle>) # dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
ex_neq_ret: assumes "a \<noteq> Return" shows "\<exists>t. \<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m a], Undecided\<rangle> \<Rightarrow> t"
2: fixes x :: "'b" assumes "x \<in> B" shows "x \<in> (\<lambda>h. h a) ` ({a} \<rightarrow>\<^sub>E B)"
IH: fixes t' :: "('sequent \<times> 'rule) tree" assumes "t' |\<in>| cont t"   and "local.wf t'" shows "ssat (fst (tree.root t'))"
Bad_def: shows "Bad = (\<lambda>i. {x \<in> space M. \<forall>N\<ge>i. real (card {n \<in> {..<N}. x \<in> Pg n}) \<le> (1 - d) * real N})"
axioms_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and r :: "'b"   and \<rho> :: "'b"   and r\<^sub>0 :: "'b"   and r\<^sub>1 :: "'b"   and s :: "'b"   and \<sigma> :: "'b"   and s\<^sub>0 :: "'b"   and s\<^sub>1 :: "'b"   and \<mu> :: "'b" assumes "arrow_of_tabulations_in_maps V H \<a> \<i> src trg r \<rho> r\<^sub>0 r\<^sub>1 s \<sigma> s\<^sub>0 s\<^sub>1 \<mu>" shows "bicategory_of_spans V H \<a> \<i> src trg"
arr_map: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and gf :: "'a \<times> 'a" assumes "hom_functor C S \<phi>"   and "partial_magma.arr (product_category.comp (dual_category.comp C) C) gf" shows "partial_magma.arr S (hom_functor.map C S \<phi> gf)"
is_NoMsg_SomeRecordExample_def: fixes r :: "'b SomeRecordExample_scheme" shows "is_NoMsg r \<equiv> Field1 r = NoMsg \<and> Field2 r = NoMsg \<and> Field3 r = NoMsg"
mem_correct_default: shows "mem_correct (\<lambda>k. State_Monad.get \<bind> (\<lambda>m. State_Monad.return (m k))) (\<lambda>k v. State_Monad.get \<bind> (\<lambda>m. State_Monad.set (m(k \<mapsto> v)))) (\<lambda>_. True)"
FF_def: shows "H.FF \<equiv> \<lambda>\<mu>\<nu>. if C.VV.arr \<mu>\<nu> then (H (fst \<mu>\<nu>), H (snd \<mu>\<nu>)) else D.VV.null"
reads_restrict1: fixes \<sigma> :: "'var \<Rightarrow> 'b"   and n :: "'n"   and \<sigma>' :: "'var \<Rightarrow> 'b" assumes "\<sigma> \<restriction> local.reads n = \<sigma>' \<restriction> local.reads n" shows "\<forall>x\<in>local.reads n. \<sigma> x = \<sigma>' x"
refine: shows "(uncurry2 remap_polys_l_impl, uncurry2 remap_polys_l2) \<in> poly_assn\<^sup>k *\<^sub>a (hs.assn string_assn)\<^sup>d *\<^sub>a polys_assn_input\<^sup>d \<rightarrow>\<^sub>a status_assn raw_string_assn \<times>\<^sub>a hs.assn string_assn \<times>\<^sub>a polys_assn"
simps_3: fixes b :: "'b set"   and c1 :: "'b com"   and c2 :: "'b com"   and s :: "'b"   and s' :: "'b" shows "Sem (Cond b c1 c2) s s' = (if s \<in> b then Sem c1 s s' else Sem c2 s s')"
independent_insertI: fixes a :: "'b"   and S :: "'b set" assumes "a \<notin> span S"   and "independent S" shows "independent (insert a S)"
False: shows "\<nexists>x. xp = \<lfloor>x\<rfloor> \<or> frs = []"
inj_iff_trivial_ker: shows "inj_on h (carrier G) = (kernel G H h = {\<one>})"