hcju/isabelleps huggingface dataset:
exhaust: fixes y :: "'a vertex" obtains x1 :: "'a" where "y = Vertex x1"   | x21 :: "'a" and x22 :: "'a" where "y = Edge x21 x22"
Pi_anti_mono: fixes A' :: "'g set"   and A :: "'g set"   and B :: "'g \<Rightarrow> 'h set" assumes "A' \<subseteq> A" shows "Pi A B \<subseteq> Pi A' B"
cases: fixes x :: "('o, 'a) arr" obtains A :: "'o" and uu_ :: "'o" and uv_ :: "'a" where "x = MkArr A uu_ uv_"   | "x = Null"
inv_solve_right: fixes G :: "('b, 'c) monoid_scheme"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "Group.group G"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "(a = b \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> c) = (b = a \<otimes>\<^bsub>G\<^esub> c)"
sup_eq_bot_iff: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and bot :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.bounded_semilattice_sup_bot sup less_eq less bot" shows "(sup x y = bot) = (x = bot \<and> y = bot)"
mHom_mem: fixes M :: "('h, 'i, 'j) Module_scheme"   and R :: "('i, 'k) Ring_scheme"   and N :: "('l, 'i, 'm) Module_scheme"   and f :: "'h \<Rightarrow> 'l"   and m :: "'h" assumes "R module M"   and "R module N"   and "f \<in> mHom R M N"   and "m \<in> carrier M" shows "f m \<in> carrier N"
comp_ide_self: fixes a :: "'b" assumes "J0.ide a" shows "J0 a a = a"
delete_invalid_block_eq: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and b :: "'b set" assumes "incidence_system point_set block_collection"   and "b \<notin># block_collection" shows "incidence_system.del_block block_collection b = block_collection"
intro: fixes R :: "('a, 'b) ring_scheme"   and S :: "('c, 'd) ring_scheme"   and h :: "'a \<Rightarrow> 'c" assumes "ring_hom_cring R S h"   and "UP_cring R" shows "UP_pre_univ_prop R S h"
extremum_strict: fixes a :: "'b multiset" shows "\<not> a \<subset># {#}"
chine_hcomp_props_3: fixes \<mu> :: "'a arrow_of_spans_data"   and \<nu> :: "'a arrow_of_spans_data" assumes "arr \<mu>"   and "arr \<nu>"   and "src \<nu> = trg \<mu>" shows "C.commutative_square \<p>\<^sub>1[Leg0 (Cod \<nu>), Leg1 (Cod \<mu>)] (Chn \<nu>) (chine_hcomp \<nu> \<mu>) \<p>\<^sub>1[Leg0 (Dom \<nu>), Leg1 (Dom \<mu>)]"
elims_3: fixes x :: "'a Vector"   and xa :: "'a Vector" assumes "\<not> x \<bottom> xa" obtains u :: "'a Vector" and v :: "'a Vector" where "x = u" and "xa = v" and "(u dot v) \<noteq> (0::'a)"
Legendre_neg: fixes a :: "int"   and p :: "int" shows "Legendre a (- p) = Legendre a p"
simps: fixes a1 :: "'b literal set set"   and a2 :: "'b literal set" shows "(a1 \<turnstile> a2) = ((\<exists>C S. a1 = S \<and> a2 = C \<and> C \<in> S) \<or> (\<exists>S C D k. a1 = S \<and> a2 = C - {k\<^sup>+} \<union> (D - {k\<inverse>}) \<and> (S \<turnstile> C) \<and> (S \<turnstile> D) \<and> k\<^sup>+ \<in> C \<and> k\<inverse> \<in> D))"
nonneg_lincomb_list_def: fixes c :: "nat \<Rightarrow> 'a"   and Vs :: "'a vec list"   and b :: "'a vec" shows "nonneg_lincomb_list c Vs b = (lincomb_list c Vs = b \<and> (\<forall>i<length Vs. (0::'a) \<le> c i))"
periodic_arithmetic_def: fixes f :: "nat \<Rightarrow> complex"   and k :: "nat" shows "periodic_arithmetic f k = (\<forall>n. f (n + k) = f n)"
times_monomial_left: fixes t :: "'a"   and c :: "'b"   and p :: "'a \<Rightarrow>\<^sub>0 'b" shows "monomial c t * p = punit.monom_mult c t p"
assms: shows "set \<Gamma> \<subseteq> set \<Gamma>'"
thetaZOCh_Sretr: fixes ch :: "'choice"   and c1 :: "('test, 'atom, 'choice) cmd"   and d1 :: "('test, 'atom, 'choice) cmd"   and c2 :: "('test, 'atom, 'choice) cmd"   and d2 :: "('test, 'atom, 'choice) cmd" assumes "compatCh ch"   and "c1 \<approx>01 d1"   and "c2 \<approx>01 d2" shows "thetaZOCh ch c1 c2 d1 d2 \<subseteq> Sretr (thetaZOCh ch c1 c2 d1 d2 \<union> ZObis)"
preserves_reflects_arr: fixes f :: "'c" shows "S.arr (\<Phi>.map f) = Cop.arr f"
prefixes_at_Suc': fixes t :: "nat"   and j :: "nat"   and b :: "nat" assumes "local.prefixes t j \<down>= b" shows "local.prefixes (Suc t) j = nxt b"
subseqs_foldr: fixes base :: "int \<times> int poly list"   and elements :: "int poly list"   and n :: "nat"   and out :: "(int \<times> int poly list) list"   and state :: "'state" assumes "subseqs_foldr sl_impl base elements n = (out, state)" shows "sli base elements n state \<and> set out = S base elements n"
polyfun_evaluate_net: fixes inputs :: "real vec list"   and m :: "(nat \<times> nat) convnet"   and j :: "nat"   and s :: "bool" assumes "map dim_vec inputs = input_sizes m"   and "valid_net m"   and "j < output_size m" shows "polyfun {..<count_weights s m} (\<lambda>f. evaluate_net (insert_weights s m f) inputs $ j)"
acquired_take_drop: fixes pending_write :: "bool"   and P :: "'b memref \<Rightarrow> bool"   and xs :: "'b memref list"   and A :: "nat set" shows "acquired (pending_write \<or> outstanding_refs is_volatile_Write\<^sub>s\<^sub>b (takeWhile P xs) \<noteq> {}) (dropWhile P xs) (acquired pending_write (takeWhile P xs) A) = acquired pending_write xs A"
leq_refl: fixes x :: "'d set" shows "x \<subseteq> x"
vertex_def: fixes G :: "('a, 'b) graph_scheme"   and x :: "'a" shows "vertex G x = (Domainp (edge G) x \<or> Rangep (edge G) x)"
at_lst_asgd_ge: fixes x :: "'a"   and y :: "'a"   and B :: "'a set"   and Pr :: "('a \<times> 'a) set" assumes "x \<in> at_least_as_good y B Pr" shows "x \<succeq>[Pr] y"
state_is_associated_string: fixes i :: "'a \<times> nat" assumes "is_valid_state_id i" shows "is_certified_associated_string (set (received_messages i)) (state i)"
linear: fixes less_eq :: "'r \<Rightarrow> 'r \<Rightarrow> bool"   and less :: "'r \<Rightarrow> 'r \<Rightarrow> bool"   and x :: "'r"   and y :: "'r" assumes "class.linorder less_eq less" shows "less_eq x y \<or> less_eq y x"
divide_eq_eq: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> '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 b :: "'c"   and c :: "'c"   and a :: "'c" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "(divide b c = a) = (if c \<noteq> zero then b = times a c else a = zero)"
mset_set_set_mset_msubset: fixes A :: "'k multiset" shows "mset_set (set_mset A) \<subseteq># A"
closed_map_restrict: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and T' :: "'d set"   and U :: "'c set"   and T :: "'d set" assumes "closedin (top_of_set (S \<inter> f -` T')) U"   and "\<And>U. closedin (top_of_set S) U \<Longrightarrow> closedin (top_of_set T) (f ` U)"   and "T' \<subseteq> T" shows "closedin (top_of_set T') (f ` U)"
group_type_def: shows "group_type \<equiv> {(g, groups_of_size g) |g. g \<in> group_sizes}"
E1: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> \<diamond>(F \<or> G) = (\<diamond>F \<or> \<diamond>G)"
eq_G: shows "mod_type_class.from_nat (i - 1) = (GREATEST n. \<not> is_zero_row_upt_k n k A)"
IH: fixes a :: "'a list"   and b :: "'b list" assumes "n < length a"   and "n < length b" shows "map2 f a b ! n = f (a ! n) (b ! n)"
f1: shows "ts' = ccs @ ccsa \<and> ct_list_eq ccsa ts"
wellformed1_alternative_sorted: fixes p :: "('a, 'b) Combinators list"   and l :: "'a set list" assumes "wellformed_policy1_strong p" shows "wellformed_policy1_strong (FWNormalisationCore.sort p l)"
beta1_def: shows "beta1 \<equiv> $pc1 = #b \<and> pc1$ = #g \<and> x$ = Suc<$x> \<and> unchanged (y, sem, pc2)"
compare_1_rat_def: fixes rai :: "int poly \<times> rat \<times> rat" shows "compare_1_rat rai = (let p = poly_real_alg_1 rai in if degree p = 1 then let x = Rat.Fract (- coeff p 0) (coeff p 1) in (\<lambda>y. compare y x) else (\<lambda>y. compare_rat_1 y rai))"
induced_induce: fixes vs :: "'a set" assumes "vs \<subseteq> verts G" shows "induced_subgraph (G \<restriction> vs) G"
set_filter_all_possible_assignments_true_is: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and s :: "'c \<times> 'd \<Rightarrow> bool option" assumes "is_valid_problem_sas_plus \<Psi>" shows "set (filter (\<lambda>(v, a). s (v, a) = Some True) (all_possible_assignments_for \<Psi>)) = (\<Union>v\<in>set (\<Psi>\<^sub>\<V>\<^sub>+). Pair v ` {a \<in> \<R>\<^sub>+ \<Psi> v. s (v, a) = Some True})"
EndPhase2_HInv3_q: fixes s :: "state"   and s' :: "state"   and q :: "Proc"   and p :: "Proc"   and d :: "Disk" assumes "EndPhase2 s s' q"   and "HInv3_L s' p q d" shows "HInv3_R s' p q d"
ProjInd_mem_eq: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and f :: "('c \<Rightarrow> 'd) \<Rightarrow> 'f"   and B :: "'f set"   and S :: "('g, 'h) aGroup_scheme"   and x :: "'f"   and y :: "'f" assumes "\<forall>j\<in>I. aGroup (A j)"   and "f \<in> carrier (a\<Pi>\<^bsub>I\<^esub> A) \<rightarrow> B"   and "bij_to f (carrier (a\<Pi>\<^bsub>I\<^esub> A)) B"   and "aGroup S"   and "x \<in> carrier (Ag_ind (a\<Pi>\<^bsub>I\<^esub> A) f)"   and "y \<in> carrier (Ag_ind (a\<Pi>\<^bsub>I\<^esub> A) f)"   and "\<forall>j\<in>I. ProjInd I A f j x = ProjInd I A f j y" shows "x = y"
non0hd: fixes x :: "'a" assumes "norm x = 1"   and "x \<in> S" shows "(0::'a) \<notin> closed_segment (h x) (- d)"
pOp_assocTr43: fixes A :: "('c, 'd) aGroup_scheme"   and a :: "'c"   and b :: "'c"   and c :: "'c"   and d :: "'c" assumes "aGroup A"   and "a \<in> carrier A"   and "b \<in> carrier A"   and "c \<in> carrier A"   and "d \<in> carrier A" shows "a \<plusminus>\<^bsub>A\<^esub> b \<plusminus>\<^bsub>A\<^esub> (c \<plusminus>\<^bsub>A\<^esub> d) = a \<plusminus>\<^bsub>A\<^esub> (b \<plusminus>\<^bsub>A\<^esub> c) \<plusminus>\<^bsub>A\<^esub> d"
f2: shows "(r'' |\<^bsub>rel\<^esub> s'') \<otimes>\<^bsub>rec_rng_of_frac\<^esub> (r |\<^bsub>rel\<^esub> s) = (r'' \<otimes> r |\<^bsub>rel\<^esub> s'' \<otimes> s)"
fsubsetI: fixes A :: "'a fset"   and B :: "'a fset" assumes "\<And>x. x |\<in>| A \<Longrightarrow> x |\<in>| B" shows "A |\<subseteq>| B"
valid: shows "valid_unMultigraph (del_unEdge v w u G)"
simplefact: fixes i :: "nat" assumes "i + 1 < length \<GG>" shows "simple_group (G\<lparr>carrier := \<GG> ! (i + 1)\<rparr> Mod \<GG> ! i)"
id_nec2_2: fixes v :: "i"   and \<alpha> :: "'b"   and \<beta> :: "'b" shows "[\<alpha> \<^bold>\<noteq> \<beta> \<^bold>\<equiv> \<^bold>\<box>\<alpha> \<^bold>\<noteq> \<beta> in v]"
coclop_coclosure: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "coclop f" shows "(f x = x) = (x \<in> range f)"
simps_2: fixes b :: "'a"   and bs :: "'a list" shows "hideDups (b # bs) = Some b # hideDupsRec b bs"
intro: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and s :: "nat"   and t :: "nat"   and cf :: "nat \<times> nat \<Rightarrow> 'b" assumes "\<exists>f. NFlow c s t f \<and> cf = residualGraph c f" shows "RGraph_axioms c s t cf"
cSup_singleton: fixes x :: "'c" shows "Sup {x} = x"
risk_neutral_prob_def: fixes M :: "'c measure"   and F :: "nat \<Rightarrow> 'c measure"   and Mkt :: "('c, 'd) discrete_market"   and r :: "real"   and risk_free_asset :: "'d"   and N :: "'c measure" assumes "rfr_disc_equity_market M F Mkt r risk_free_asset" shows "rfr_disc_equity_market.risk_neutral_prob F Mkt r N = (prob_space N \<and> (\<forall>asset\<in>stocks Mkt. martingale N F (discounted_value r (prices Mkt asset))))"
iso_\<eta>': shows "D.iso (\<Phi> (g, f) \<cdot>\<^sub>D (F g \<star>\<^sub>D F f) \<cdot>\<^sub>D (\<mu> \<star>\<^sub>D F f) \<cdot>\<^sub>D \<phi> \<cdot>\<^sub>D D.inv (unit (src\<^sub>C f)))"
L1: shows "nat_to_pr (s_ce e x) (c_pair y z) = 0"
1: shows "fmrestrict_set ((vsa \<union> vsb) \<inter> vs) (fma ++ fmb) = fmrestrict_set ((vsa \<union> vsb) \<inter> vs) fmb ++\<^sub>f fmrestrict_set ((vsa \<union> vsb) \<inter> vs) fma"
h_4: shows "trpl_code (0, Bk \<up> m, Oc \<up> rs @ Bk \<up> n) = conf (code tp) (bl2wc (<lm>)) stp"
SUP_Lim: fixes X :: "nat \<Rightarrow> 'a"   and l :: "'a" assumes "incseq X"   and "X \<longlonglongrightarrow> l" shows "Sup (range X) = l"
ground_derive_chain: assumes "chain (\<leadsto>) Sts" shows "chain sr_ext.derive (lmap (\<lambda>St. grounding_of_clss (N_of_state St \<union> P_of_state St \<union> Q_of_state St)) Sts)"
extract_viewres_a_def: fixes r :: "('a, 'b) viewres" shows "extract_viewres_a r \<equiv> case Rep_viewres r of Some (a, t) \<Rightarrow> a"
Thabit_ibn_Qurra_Rule_Amicable: fixes k :: "nat"   and p :: "nat"   and q :: "nat"   and r :: "nat" assumes "1 < k"   and "prime p"   and "prime q"   and "prime r"   and "p = 2 ^ (k - 1) * 3 - 1"   and "q = 2 ^ k * 3 - 1"   and "r = 2 ^ (2 * k - 1) * 9 - 1" shows "(2 ^ k * p * q) Amic (2 ^ k * r)"
sym_in_ground_heads: fixes \<zeta> :: "('s, 'v) hd" assumes "is_Sym \<zeta>" shows "hd.sym \<zeta> \<in> local.ground_heads \<zeta>"
cont_SUP: fixes lub :: "'d set \<Rightarrow> 'd"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e set"   and g :: "'d \<Rightarrow> 'e \<Rightarrow> 'a" assumes "mcont lub ord \<Union> (\<subseteq>) f"   and "\<And>y. mcont lub ord Sup (\<le>) (\<lambda>x. g x y)" shows "cont lub ord Sup (\<le>) (\<lambda>x. \<Squnion> (g x ` f x))"
exhaust: fixes y :: "'e\<cdot>'f error" obtains "y = \<bottom>"   | e :: "'f" where "y = Err\<cdot>e"   | a :: "'e" where "y = Ok\<cdot>a"
hyps_for_simp': fixes h :: "'outPort"   and n :: "'node"   and p :: "'inPort" shows "(h \<in> fset (hyps_for n p)) = (hyps n h = Some p)"
tt_min_keys: fixes v :: "'t"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "v \<in> keys p" shows "tt p \<preceq>\<^sub>t v"
norm: shows "LIM x inf F (principal {z. f z \<noteq> 0}). cmod (f x) :> at_right 0"
RootExSem_Status: fixes ST :: "('d, 'e, 'f) status" shows "RootExSem (SAs (HA ST)) (CompFun (HA ST)) (Conf ST)"
corec: fixes g1 :: "'h \<Rightarrow> 'g"   and g2 :: "'h \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> (('e \<times> 'b list) \<times> 'a) spmf"   and g3 :: "'h \<Rightarrow> 'a \<Rightarrow> 'd \<Rightarrow> (('f \<times> 'b list) \<times> 'a) spmf"   and a :: "'h" shows "corec_rest_scheme g1 g2 g3 a = Rest (g1 a) (g2 a) (g3 a)"
single_valuedp_def: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "single_valuedp r = (\<forall>x y. r x y \<longrightarrow> (\<forall>z. r x z \<longrightarrow> y = z))"
divideC_field_splits_simps_2_3: fixes c :: "complex"   and b :: "'b"   and a :: "'b" assumes "0 < c" shows "(b /\<^sub>C c \<le> a) = (if 0 < c then b \<le> c *\<^sub>C a else if c < 0 then c *\<^sub>C a \<le> b else (0::'b) \<le> a)"
less_1_mult: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and m :: "'a"   and n :: "'a" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less one m"   and "less one n" shows "less one (times m n)"
case_1: fixes f1 :: "nat \<Rightarrow> 'a"   and f2 :: "'a"   and nat :: "nat" shows "(case enat nat of enat x \<Rightarrow> f1 x | \<infinity> \<Rightarrow> f2) = f1 nat"
cs_when_recording: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and i :: "nat"   and j :: "nat"   and t :: "('d, 'e, 'f) event list"   and p :: "'d"   and cid :: "nat"   and q :: "'d" assumes "computation channel trans send recv init final"   and "i < j"   and "j \<le> length t"   and "distributed_system.trace channel trans send recv init t final"   and "ps (distributed_system.s channel trans send recv init t i) p \<noteq> None"   and "snd (cs (distributed_system.s channel trans send recv init t i) cid) = Recording"   and "snd (cs (distributed_system.s channel trans send recv init t j) cid) = Done"   and "channel cid = Some (p, q)" shows "map Msg (fst (cs (distributed_system.s channel trans send recv init t j) cid)) = map Msg (fst (cs (distributed_system.s channel trans send recv init t i) cid)) @ takeWhile ((\<noteq>) Marker) (msgs (distributed_system.s channel trans send recv init t i) cid)"
app: fixes P :: "'b \<Rightarrow> bool"   and t :: "'b"   and u :: "'b" assumes "simple_syntactic_and P" shows "P (app t u) = (P t \<and> P u)"
length_take: fixes n :: "nat"   and xs :: "'e list" shows "length (take n xs) = min (length xs) n"
rwof_init: fixes m0 :: "'a nres"   and cond :: "'a \<Rightarrow> bool"   and step :: "'a \<Rightarrow> 'a nres" assumes "nofail (m0 \<bind> WHILE cond step)" shows "m0 \<le> SPEC (rwof m0 cond step)"
\<phi>_def: shows "\<phi> = (1 + sqrt 5) / 2"
finite_imp_fold_graph: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'f"   and z :: "'f" assumes "finite A" shows "\<exists>x. fold_graph f z A x"
inject: fixes x21 :: "'a tree"   and x22 :: "'a"   and x23 :: "'a tree"   and y21 :: "'a tree"   and y22 :: "'a"   and y23 :: "'a tree" shows "(\<langle>x21, x22, x23\<rangle> = \<langle>y21, y22, y23\<rangle>) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23)"
serializable_encoding_decoded_plan_is_serializable: fixes \<Pi> :: "'a strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and t :: "nat" assumes "is_valid_problem_strips \<Pi>"   and "\<A> \<Turnstile> \<Phi>\<^sub>\<forall> \<Pi> t" shows "STRIPS_Semantics.is_serial_solution_for_problem \<Pi> (concat (\<Phi>\<inverse> \<Pi> \<A> t))"
negligible_iff_null_sets: fixes S :: "'b set" shows "negligible S = (S \<in> null_sets lebesgue)"
drop_all_conc: fixes a :: "'c list"   and b :: "'c list" shows "drop (length a) (a @ b) = b"
goedel_first_theEasyHalf_pos: assumes "prv \<phi>G" shows "prv fls"
add_rquot: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.cancellative_partial_semigroup times D"   and "partial_semigroup.gR_rel times D x y" shows "(D x z \<and> times x z = y) = (z = cancellative_partial_semigroup.rquot times D y x)"
semInt_prVar: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" assumes "wlsSEM SEM" shows "prVar (semInt SEM) SEM"
maximal_independent_subset_extend: fixes S :: "'b set"   and V :: "'b set" assumes "S \<subseteq> V"   and "independent S" obtains B :: "'b set" where "S \<subseteq> B" and "B \<subseteq> V" and "independent B" and "V \<subseteq> span B"
prod: shows "(\<Prod>a\<leftarrow>a # as. [:a, 1::'a:]) = [:a, 1::'a:] * (\<Prod>a\<leftarrow>as. [:a, 1::'a:])"
solve_factored_ratfps: fixes cs :: "('b \<times> nat) list"   and p :: "'b poly" assumes "distinct (map fst cs)"   and "(0::'b) \<notin> fst ` set cs" shows "fps_of_poly p / fps_of_poly (\<Prod>(c, n)\<leftarrow>cs. [:1::'b, - c:] ^ Suc n) = Abs_fps (interp_ratfps_solution (solve_factored_ratfps p cs))"
exhaust: fixes y :: "top0" obtains x11 :: "char list" and x12 :: "spec list option" and x13 :: "dec list" where "y = Tmod x11 x12 x13"   | x2 :: "dec" where "y = Tdec x2"
length_append_singleton: fixes xs :: "'e list"   and x :: "'e" shows "length (xs @ [x]) = Suc (length xs)"
chambercomplex_image: shows "ChamberComplex (f \<turnstile> X)"
assms: shows "\<langle>\<langle>r, A\<rangle>, s\<rangle> \<in>\<^sub>\<circ> vlrestriction D"
assms_4: fixes a :: "nat list"   and b :: "nat list"   and i :: "nat" assumes "(a, b) \<in> set subsets"   and "i \<in> set a \<or> i \<in> set b" shows "i < length qs"
grounded_res: shows "ord_resolve (S_M S M) CAs DA AAs As \<sigma> E"
simulation_steps: fixes a :: "'a"   and as :: "'a list"   and b :: "'b" assumes "A.steps (a # as)"   and "a \<sim> b"   and "PA a"   and "PB b" shows "\<exists>bs. B.steps (b # bs) \<and> list_all2 (\<lambda>a b. a \<sim> b \<and> PA a \<and> PB b) as bs"
reduction_subst: fixes t :: "trm"   and t' :: "trm"   and x :: "name"   and v :: "trm" assumes "t \<mapsto> t'" shows "t[x::=v] \<mapsto> t'[x::=v]"
EQ_And: fixes f :: "atom fm"   and g :: "atom fm"   and xs :: "'a list" shows "EQ (And f g) xs = EQ f xs \<union> EQ g xs"
active_geq_nxtAct: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and i :: "nat"   and n :: "nat" assumes "\<parallel>c\<parallel>\<^bsub>t i\<^esub>"   and "the_enat \<langle>c #\<^bsub>enat n\<^esub>inf_llist t\<rangle> \<le> the_enat \<langle>c #\<^bsub>enat i\<^esub>inf_llist t\<rangle>" shows "\<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub> \<le> i"
big_forest_path_split_disj: fixes H :: "'a"   and c :: "'a"   and a :: "'a"   and b :: "'a"   and d :: "'a" assumes "equivalence H"   and "bijective (c * top) \<and> bijective (c[[top]])"   and "regular a \<and> regular b \<and> regular c \<and> regular d \<and> regular H" shows "bf_between_arcs a b H (d \<squnion> c) = (bf_between_arcs a b H d \<or> bf_between_arcs a c H d \<and> bf_between_arcs c b H d)"
reindex: fixes h :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and g :: "'c \<Rightarrow> 'd" assumes "inj_on h A" shows "prod g (h ` A) = prod (g \<circ> h) A"
finite: shows "\<forall>ts v. nP v = ts \<longrightarrow> finite ts"
S6_excl: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> S6 rmhist p \<longrightarrow> S6 rmhist p \<and> \<not> S1 rmhist p \<and> \<not> S2 rmhist p \<and> \<not> S3 rmhist p \<and> \<not> S4 rmhist p \<and> \<not> S5 rmhist p"
triangle_in_hom_3: shows "\<guillemotleft>\<l>[f] \<cdot> (\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>) \<cdot> \<r>\<^sup>-\<^sup>1[f] : f \<Rightarrow> f\<guillemotright>"
run_alt_env: fixes alt :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and m1 :: "('b, 'a) envT"   and m2 :: "('b, 'a) envT"   and r :: "'b" shows "run_env (alt_env alt m1 m2) r = alt (run_env m1 r) (run_env m2 r)"
not_plus_prv_neg_eql_pls: fixes m :: "nat"   and n :: "nat"   and k :: "nat" assumes "m + n \<noteq> k" shows "prv (neg (eql (pls (Num m) (Num n)) (Num k)))"
simps_2: fixes l :: "linear_poly"   and r :: "QDelta"   and v :: "nat" shows "qdelta_constraint_to_atom' (GEQ_ns l r) v = Geq v r"
mono_Field: fixes r :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" assumes "r \<subseteq> s" shows "Field r \<subseteq> Field s"
V_subseteq_R1: shows "V_constotal \<subseteq> \<R>"
dgrad_setI: fixes d :: "'a \<Rightarrow> nat"   and t :: "'a"   and m :: "nat" assumes "d t \<le> m" shows "t \<in> dgrad_set d m"
cindex_pathE_constI: fixes g :: "real \<Rightarrow> complex"   and c :: "complex"   and z :: "complex" assumes "\<And>t. \<lbrakk>0 < t; t < 1\<rbrakk> \<Longrightarrow> g t = c" shows "cindex_pathE g z = 0"
simps_4: fixes ao :: "'a option"   and x :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "Compl_set_less_eq_aux ao (x # xs) (y # ys) = (if x < y then \<not> proper_interval ao (Some x) \<and> Compl_set_less_eq_aux (Some x) xs (y # ys) else if y < x then \<not> proper_interval ao (Some y) \<and> Compl_set_less_eq_aux (Some y) (x # xs) ys else \<not> proper_interval ao (Some y))"
total_derivative_ll_on_open: fixes x0 :: "'a" shows "ll_on_open (Y.existence_ivl0 x0) (\<lambda>t. (o\<^sub>L) (Y.vareq x0 t)) UNIV"
homI: fixes CC :: "('d, 'e, 'f) category_scheme"   and f :: "'e"   and A :: "'d"   and B :: "'d" assumes "category CC"   and "f \<in> Ar\<^bsub>CC\<^esub>"   and "Dom\<^bsub>CC\<^esub> f = A"   and "Cod\<^bsub>CC\<^esub> f = B" shows "f \<in> Hom\<^bsub>CC\<^esub> A B"
mult_matrix_n_zero_left: fixes fadd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and fmul :: "'d \<Rightarrow> 'c \<Rightarrow> 'b"   and n :: "nat"   and A :: "'c matrix" assumes "fadd (0::'b) (0::'b) = (0::'b)"   and "\<forall>a. fmul (0::'d) a = (0::'b)" shows "mult_matrix_n n fmul fadd 0 A = 0"